It meant United extended their winning streak to six successive matches, and nine games unbeaten.
Laws feared his side were going to be leaving the Abbey Stadium pointless, a result which would have been considered a 'shock' thanks to the 22 League places between the sides going into the game.
He said: "It was a very poor first-half in which we never got going and continually gave the ball away.
S"omething drastic had to be done which was why I made two substitutions at half time, and we started looking like the kind of team we have been during our very good run.
"Having said that though it just didn't look as though it was going to be our night. Andy Butler alone had four chances which he failed to put away when it looked more difficult not to score.
"But we kept at it, Taylor was one of the players I sent on in the second half, ripped Cambridge apart down the right wing and once we got the equaliser I knew we could win it as long as there was enough time."
After Steve Torpey's equaliser afgter 87 minutes, Taylor grabbed the winner in the second minute of stoppage time.
Cambridge's under-pressure coach Herve Renard was left gutted.
He said: "We played with great heart against the leaders, and deserved to be in front at half time. We scored a good goal when one of our new players Colin Heath set up Richard Hodgson.
"But we didn't have enough possession in the second half, Scunthorpe were able to attack all the time, and especially when you are down to 10-men against the leaders it is very difficult.
"I thought we would get something out of the game but that did not happen and now we have to try to keep our heads up."