Cambridge have now won five and drawn one of their six matches since the turn of the year, with their five wins in a row in all competitions the longest winning streak since Mark Bonner took over.
Story of the match
Bonner made three changes to the side that defeated Doncaster Rovers on Tuesday night.
Cambridge made a positive start to the match as May had a shot blocked inside the box, while Harvey Knibbs headed wide from Jack Iredale's cross in the third minute.
Oliver Finney went closest for Crewe on 12 minutes when the midfielder flashed a left-footed effort narrowly wide of the right-hand upright with an effort from inside the box.
The midfielder was picked out once again by Mikael Mandron soon after, but he was unable to get a shot away.
On the half-hour mark, Dimitar Mitov held Tom Lowery's free-kick while at the other end, Knibbs fired over from inside the box after being picked out by Williams.
Billy Sass-Davies had his shot blocked from a Luke Offord free-kick before the U's broke the deadlock.
Knibbs won back possession as he teed up May, whose effort from outside the penalty area crashed in off the underside of the bar to send Cambridge into halftime with the lead.
The hosts started the second half well, with Sam Sherring heading over from Iredale's corner a few minutes after the restart.
On 52 minutes, Chris Long latched onto a high ball - but he couldn’t steer it towards goal with his first touch.
Crewe had a good spell in the 64th minute when Long and Tom Lavery had shots blocked, while Mitov did well to prevent a ricochet off Sherring going past him.
Zac Williams then headed wide from a free-kick, with Smith missing the target at the other end after being played in by James Brophy with quarter of an hour to go.
The Railwaymen pushed for an equaliser, but could rarely test Mitov, with the 'keeper saving Long's effort from outside the box.
Substitute Luke Murphy lashed a volley wide deep into stoppage time – it proved to be the last action as the hosts persevered for yet another three points.
Man of the match: Adam May
He scored the only goal of the match to give Cambrdige three crucial points to keep their momentum going.