In front of an excellent sun-bathed crowd of 2,239, the U`s just about deserved the points against a home side still smarting from Saturday`s hammering at Dagenham & Redbridge.
The only goal of the game came after 23 minutes when Craig Eastmond collected a Josh Taylor pass and rifled the ball into the top corner past David Gregory.
Josh Rees had an effort from 25 yards go just over for Bromley, while Taylor came close to doubling the visitors` lead 11 minutes from time, but his effort curled just wide of the mark.
Five minutes before the end, Frankie Raymond had the Sutton supporters` hearts in their mouths as his powerful header just cleared the bar.
Dagenham lost top spot and their scoring boots as a Maidstone United side who beat Sutton on Saturday held them to a 0-0 draw at the Gallagher Stadium.
It was an entertaining encounter, despite the lack of goals and there were good chances at both ends, but no goals.
The Daggers finished a man down after right-back Sam Ling picked up two yellow cards in a minute around the 73rd minute-mark.
Just booked for a foul on Sam-Yorke, he went in on Jai Reason and gave the referee little option.
But they held firm for the final 17 minutes or so to maintain their unbeaten record.
Leyton Orient moved up to third in the early season table after a hard-fought 1-0 win at newly-promoted Maidenhead United.
Alan Devonshire`s hosts had earned a point from their long trip to Cumbria against Barrow AFC on Saturday, but found the O`s just a little too good for them on the day at York Street.
Macauley Bonne’s early 6th minute header proved to be the difference on a sweltering afternoon, guiding a James Dayton cross beyond Carl Pentney in the Magpies goal.
Maidenhead came close to grabbing a point late on as they threw bodies forward in search of an equaliser and sub Sam Barratt first headed wide before seeing an ambitious shot from the halfway line drop just over Charlie Grainger`s the bar.
Boreham Wood saw their decent start to the new season falter as Wrexham moved up to fourth after they too won 1-0 at Meadow Park.
And it was in-form loanee striker Alex Reid who scored his 3rd goal in four games when he headed the winner from a Paul Rutherford corner just after the hour-mark.
Kieran Murtagh came closest for the home side but saw his shot saved by Christian Dibble.
And Ntumba Masanga forced a fine save from Grant Smith at the other end.
The pressure was being heaped on Hartlepool United manager Craig Harrison after his relegated side had failed to find a win in their first six matches.
Saturday`s loss at home to newly-promoted AFC Fylde was immensely disappointing, so today`s game at Nethermoor against fellow strugglers Guiseley AFC was crucial.
And it was joy for Harrison and Pools as they picked up their first victory of the season with a 1-0 win.
Jonathan Franks scored the only goal of the game, 4 minutes into the second half - his first since re-joining for the club.
Franks netted from close range, but they visitors came agonisingly close to falling behind as early as the 4th minute, but Frank Mulhern`s effort crashed against the underside of the bar and out again.
The first half this afternoon was lacking in entertainment but did see Pools go in at half time on level terms for the first time this season.
And then Pools `keeper Scott Loach pulled off a brilliant double save to deny Danny Lowe and the follow-up header from sub Kayode Odejayi late on.
Chester FC just can`t seem to win at home.
And in front of a holiday crowd of 2,363, Jon McCarthy`s side suffered another loss on home soil against Cheshire rivals Macclesfield Town.
Looking to back up Saturday`s excellent win at Aldershot Town, Chester fans came to the game in hope that their home hoodoo would come to an end.
But goals from Mitch Hancox after just 7 minutes and then Kieran Kennedy on 62 minutes gave John Askey’s side the points.
Woking were certainly not in confident mood when they took on struggling Torquay United at Kingfield.
The Cards had suffered successive defeats, but came up against a managerless Gulls outfit completely shorn of any confidence at all and another heavy, 4-1 defeat means it six straight losses for the Devonians and a new manager is now urgently required.
They conceded two goals in 10 minutes to Woking's Joe Ward (23 minutes) and winger Jason Banton.
Torquay pulled one back on the stroke of half-time, when James Gray scored a goal-of-the-season contender to pull the score back to 2-1.
And they were then blown away when goals by Charlie Carter and Bobson Bawling in the 51st and 58th minutes ended the game as a contest.
The Kent derby between Dover Athletic and newly-promoted Ebbsfleet United ended all-square at Crabble.
Ryan Bird gave the home side a 17th minute lead.
But Darren McQueen finally managed to score the equaliser 14 minutes from time after they had dominated the second half.
And a sixth draw from seven matches means that Darryl McMahon`s side are, along with Dagenham, the only other side het to taste defeat in 2017/18.
Tranmere Rovers managed a second win of the campaign at Solihull Moors, who now find themselves second-bottom, despite Saturday`s win at Torquay.
One of Micky Mellon`s close season signings, Ollie Norburn, gave Rovers a 19th minute lead, showing coolness in front of goal before beating goalscoring `keeper Nathan Vaughan.
And they wrapped up victory 8 minutes from time when striker Andy Cook got on the end of a Connor Jennings cross to beat Vaughan.
Ten-man FC Halifax Town held former boss Neil Aspin’s Gateshead to a goalless draw at the International Stadium.
Matty Kosylo, who scored the opener in front of the BT Sport cameras on Saturday, saw red on 66 minutes for a second yellow card having hit the woodwork earlier in the match.
The Shaymen`s Adam Morgan also hit the post as the newly-promoted outfit held on for a good point
It also finished 0-0 at the Silverlake Stadium between Eastleigh and Aldershot.
But that doesn`t tell half the story as The Shots had to battle for almost an hour with ten-men following the 36th minute dismissal of Kodi Lyons-Foster.
Lyons-Foster stretched for a tackle with Sam Wood and was shown a straight red card as the referee deemed it dangerous and reckless.
A backs-to-the-wall second half performance earned Gary Waddock`s side a share of the holiday spoils, much to the joy of a healthy number of visiting fans in the 2,525 crowd.