In a first half, the hosts took the game to the U's and went ahead through striker Chris Dagnall.
Further bad news in that first period came early, when Sammie Szmodics went down under challenge and couldn't continue.
After the break, the U's threw everything at Crewe, but were hit on the break and Dagnall doubled the lead.
He inspected the pitch on two occasions, and finally gave the go-ahead thirty minutes before three o'clock.
There were doubts about whether the game would take place, with flooded areas on the surface troubling the referee.
Dickenson and Fosu were both denied by saves in injury time, but it remained 2-0 to the hosts.
The home team started the game, which kicked off fifteen minutes later than planned, on the front foot and there was only two minutes on the clock when Sam Walker palmed away a long range shot.
Three minutes later, and Jordan Bowery got away from George Elokobi but shot well wide from the edge of the box.
The on loan forward then had a much better chance, but volleyed over from a great knockdown from George Ray.
U’s forward Sammie Szmodics then went down in a challenge and looked to be holding his left ankle before being helped off the pitch.
He wasn't able to continue, and Tarique Fosu entered the fray and took up Szmodics' position just behind Rekeil Pyke and Chris Porter.
The U's first effort on goal came on sixteen minutes, but Sean Murray's shot from distance flew well wide.
Crewe were having the better of things as the first half progressed, and it was no surprise when they went ahead on twenty five minutes.
A wide free kick was whipped into the near post and dangerman Chris Dagnall got there first and his flicked header could only be helped into the net by Walker's right hand.
Sub Fosu came closest for the U's in the remainder of the first period, but his acrobatic volley went just the wrong side of the post.
That chance on the half hour was the final goalmouth action before the break, but it was a half that the hosts dominated and were deservedly a goal ahead.
Whatever John McGreal said to his U's side, it had the desired effect and it was the visitors who started the second half best.
Sean Murray was first to try his luck, but his shot from just outside the box went wide of the target.
Murray then capitalised on a great flick from Pyke to break, and the former Watford forward played in Fosu, who blazed over the bar with his right foot shot.
The best chance of the day for the U's fell to loanee Pyke, but his far post header was blocked by Garratt.
Pyke and Fosu then combined on the right and when the latter crossed, Porter's header was comfortably saved by Garratt.
It was all U's at this stage, and Garratt had more work to do when he fielded a Fosu free kick just after the hour mark.
Another U's attack saw Fosu shoot across goal, after great work on the left by Matt Briggs and Pyke.
A rare attack for the hosts came about when Tom Lapslie lost possession in midfield, but the subsequent shot went wide of the target.
Porter once again won his header when a cross came into the box, but his looping header wasn't strong enough to beat Garratt in the home goal.
With six minutes remaining, Lapslie was fouled in midfield but the referee decided that an advantage could be had.
There was no advantage though, and Crewe broke to make the game safe for the home side when Dagnall slid in a cross to double his tally.
He was denied a hat-trick with two minutes of the ninety remaining, with Walker blocking the close range effort from another effective low cross.
Garratt made two top drawer saves in injury time, from Dickenson and Fosu, and ensured that his side held out for the three points.
The U's return to the Weston Homes Community Stadium in seven days’ time, with fellow promotion chasers Luton Town visiting north Essex.