A blustery evening in North Wales saw Solihull cause the home side problems, especially in the second half, and created several opportunities, but found Chris Dunn in the Wrexham goal capable of dealing with everything thrown at him.
What turned out to be the only goal of the game came 6 minutes before the interval when Paul Rutherford hooked the ball past Nick Townsend in the Moors` goal from close range.
John Still`s Daggers side moved up to fifth after beating overnight leaders Dover at Victoria Road.
The Whites` unbeaten run of ten games was ended when a header after 23 minutes from Morgan Ferrier from a Sam Ling cross eventually proved enough to win the game for the hosts.
Home goalkeeper Mark Cousins made good saves from Mitch Pinnock and James Alabi, while at the other end, Corey Whitely hit the bar 16 minutes from time.
Macclesfield Town are now third, but they could only manage a 2-2 Moss Rose draw with Tranmere Rovers, who are now fourteenth.
The Silkmen still moved up a place in the table due to Sutton United`s surprise home defeat to strugglers Torquay United.
It was underachieving Tranmere who took the lead after 19 minutes when former Manchester United youngster Larnell Cole broke away to net past Shwan Jalal.
Five minutes before half-time, Macclesfield pulled back on level terms when they were awarded a penalty. Scott Davies saved Danny Whitaker`s spot-kick, only for Scott Wilson to pounce on the loose ball to equalise.
However, just 2 minutes later, Tranmere were back in front when Andy Cook converted well inside the box after some good play out wide from James Norwood.
The second half saw Micky Mellon`s side manage the game well and they looked on course for three vital points.
But in the 2nd minute of stoppage time at the end of the game, sub Tyrone Marsh turned the ball past Davies at the far post.
Even then, Tranmere came close to winning it when they immediately went up the other end and a Norwood effort from 25 yards forced a decent save out of Jalal.
Torquay moved off the foot of the table after their 1-0 win at Gander Green Lane.
Sutton came into the game knowing a win could take them to the top of the table if results went their way.
However, they were denied a third successive league win by a determined Gulls side being re-built by recently-appointed boss Gary Owers.
Forward Ruairi Keating, who hadn`t scored in the league since August, ended his goal drought when he gave Torquay a 10th minute lead, heading in after a flick-on from Jamie Reid.
And far from simply holding on to that lead after that, Torquay took the game to the U`s and forced home `keeper Jamie Butler into making two or three decent saves.
At the other end, visiting number one, Vincent Dorel, was also called into action in the second half as Sutton poured forward in search of a leveller that never came.
Maidstone United occupy the final play-off spot, but suffered a 4-0 drubbing in front of their own fans at the Gallagher Stadium against in-form Boreham Wood.
Wood came into the game on the back of an excellent run of form which included their terrific win against League One Blackpool in the Emirates FA Cup.
And it took them only 13 minutes to make the breakthrough, albeit in rather fortuitous circumstances when Kane Smith`s cross drifted over United `keeper Lee Worgan and into the back of the net.
Five minutes before the break, Dan Holman doubled the visitors lead, and then 8 minutes into the second period, the loanee striker drove low past Worgan to make it 3-0.
Then, in the 77th minute, attacking midfielder Blair Turgott broke into the box and was hauled down, with the referee pointing straight to the spot.
Turgott dusted himself down before stepping up and converting the resulting penalty confidently past Worgan.
After successive league draws, Woking recorded three points on Tuesday to stay in top seven contention after a 1-0 win against Ebbsfleet United at Kingfield.
What proved to be the only goal of the game arrived on the half-hour mark when Inih Effiong controlled a ball over the top perfectly before slotting it past Nathan Ashmore.
Kane Ferdinand had a couple of opportunities to add to the Cards` lead, while at the other end, keeper Sam Mason saved well from Danny Kedwell.
But the home side held firm to inflict a fifth defeat on Fleet.
Hartlepool United, beaten at home by Aldershot Town last weekend, returned to winning ways on Tuesday at the Victoria Ground when easing past out-of-form FC Halifax Town.
Pools raced out of the traps and were 2-0 up inside the opening 7 minutes.
The opener came after just 4 minutes when Nicky Deverdics played a one-two with Michael Woods on the edge of the area before beating Sam Johnson.
And the lead was quickly doubled when Rhys Oates was bundled over in rhe area and Liam Donnelly stepped up to convert the penalty.
Then, on the half-hour mark, Pools added a third when Oates fired past Johnson after good work from Michael Woods.
The Shaymen offered up little threat themselves, and a quieter second period finally found another goal in the final minute of normal time when Oates and Woods combined to set up Conor Newton to make it 4-0.
Another game in the north-east saw Gateshead`s push towards the play-off places take a blow when they were beaten 2-1 at the International Stadium by Barrow AFC, whose fall towards the bottom of the table was halted after recording back-to-back wins.
Byron Harrison, who has found goals harder to find this season, gave the Bluebirds a 33rd minute lead with a hard, low shot from just inside the area.
However, Gateshead pulled themselves back on level terms 2 minutes before the break when Wes York got on the end of a good Luke Hannant pass to beat Stuart Moore.
What turned out to be the winning goal came on the hour-mark – and it was a goal worthy of winning any match when Harry Panayiotou stepped up and fired a 30-yard effort past a stunned Dan Hanford.
Two sides not exactly in the best of form at present, Leyton Orient and Chester FC, shared a 2-2 draw at the Matchroom Stadium, leaving both in twentieth and twenty-first places respectively.
The O`s, who hadn`t won in seven in all competitions, will be much the happier with a point in the end as they played the entire second half with only ten-men.
Macauley Bonne headed managerless O`s into an early 6th minute lead.
James Akintunde pulled the visitors back on level terms 9 minutes later when his effort was deemed to have crossed the line.
A free-kick converted by Ross Hannah then put Chester in front in the 40th minute.
And then, a minute before the break, Nyal Bell capitalises on a defensive mix-up and went down under Jamie Sendles-White's challenge.
Sendles-White was subsequently shown his second yellow card and dismissed, but Hannah`s penalty was saved by Sam Sargeant.
The referee made himself even more unpopular with home fans when he sent caretaker boss Ross Embleton into the stands.
However, the ten-man performed heroically and on the hour-mark, Orient equalised with a stunning goal from the experienced Jobi McAnuff, whose brilliant strike from 30-yards left Chester `keeper Alex Lynch rooted to the spot.
Another side currently struggling for form, Eastleigh, fought back to earn a point from another 2-2 draw with Maidenhead United at the Silverlake Stadium.
The Magpies broke the deadlock 4 minutes before half-time through Harold Odametey.
And then 6 minutes into the second half, Maidenhead were awarded a penalty after Sam Barratt was tripped in the area and Harry Pritchard duly converted.
However, it was Pritchard who inadvertedly began the Spitfires comeback when he conceded an 84th minute penalty which Ayo Obileye netted.
Three minutes later, Maidenhead lost James Mulley to a straight red card for a rash challenge and 3 minutes into stoppage time, Ben Williamson equalised with a fine 25-yard strike.
Two of Tuesday's games were postponed.
The match between Bromley and Aldershot Town was postponed 10 minutes before kick-off because of an electrical fault at Hayes Lane, while Mill Farm was waterlogged, causing AFC Fylde to call off their game against Guiseley AFC.