Spenny picked up their fourth win of the season, their second away from home to lift them into fourth - watch match highlights here.
A brace from Glen Taylor and a late strike from Andrew Johnson secured the points for the visitors, in what was the only fixture across the National League after the original match was postponed.
Boston have picked up six points from their opening five matches but didn't reach the performance levels of the weekend when they beat Chorley.
"It's tough to take when we put in such a good performance on Saturday, it was so accomplished and controlled and then for 45 minutes tonight we looked like we hadn't ever played with each other," said manager Murray.
"At times in the first-half it looked like men vs boys. Our first 45 minutes was very naive, we didn't control the game, we didn't deal with problems early enough and they became issues.
"We made some changes at half-time and they got us back into game. We created chance after chance after chance and if we take one of those then it's a different game."
Murray gave plenty of credit to Spennymoor, saying: "It was an opponent that was on another level to anything we've faced so far. They showed that they've been together for a while and that they had a real belief and confidence.
"They were machine-like in the patterns of play that they had. That comes from having a team together for a year or two. We're the opposite end of that at the minute. We're still at the start of our journey, learning and having to take punches on the chin."