The home side were some way short of their best, but they did enough to set up a meeting with Sheffield United next month.
Constable opened the scoring with his seventh goal of the season, heading in Leven's 67th-minute corner.
The Scot made the game safe ten minutes from time with a terrific volley and the closing stages were further boosted by the welcome return of Michael Duberry.
It stretched United's unbeaten run to seven games and gave the side a welcome lift heading into the Christmas fixtures.
Liam Davis replaced Tony Capaldi in the only change to United's starting line-up from the 1-1 draw at Morecambe on Saturday.
More significantly, Duberry was named among the substitutes for the first time this season.
The tie, not helped by the timing so close to the festive period, did not attract a big crowd, with Accrington reportedly selling just six tickets in advance.
It meant there was a palpable lack of atmosphere at kick-off, which may have contributed to a first half low on quality.
United started slowly, while Accrington's final ball was incredibly wasteful.
The first noteworthy incident came on 16 minutes, when Lee Cox came on for a limping Adam Chapman.
Midway through the half the U's threatened to spark into life.
They were unlucky not to take the lead, but Stanley defender Aristote Nsiala cleared two efforts off the line in the space of 30 seconds.
Damian Batt surged forward down the right flank and swung in an excellent cross for Constable, who controlled well before seeing his shot turned behind.
From the resulting corner the ball fell to Michael Raynes with his back to goal.
The centre back had scored a late equaliser in the first game to set up the replay and he came close to another goal with an improvised shot over his shoulder which Nsiala headed to safety.
Batt, who was excellent at Morecambe, threatened again shortly afterwards, but his cross was headed wide by Sean Rigg.
Accrington forced several corners, but U's goalkeeper Ryan Clarke only had a handful of crosses to deal with before the break.
At the other end his opposite number Paul Rachubka was forced into action late in the half to keep out a low shot from Constable.
Stanley looked to freshen the game up, bringing on Padraig Amond and Bohan Dixon early in the second half.
The game belatedly came to life around the hour mark.
It was sparked by Luke Clark, who stopped a promising United attack when he went down clutching his head.
The referee had no option but to halt the game, but the home crowd were convinced the midfielder had made the most of the it.
It gave the game an edge and United suddenly looked sharper.
Rigg had a penalty appeal turned down, but midway through the half the deadlock was finally broken.
Potter intercepted a loose ball in midfield and drove forward. He slipped in Constable, who cut inside his marker and fired in a shot which Rachubka shovelled round the post.
From the resulting corner the striker leapt to nod Leven's delivery into the net.
United almost shot themselves in the foot ten minutes later.
Clarke fumbled Lee Molyneux's free-kick and fell backwards, but despite the visitors' appeals it had crossed the line, the referee was not interested.
Within three minutes United had sealed victory.
Leven, who has had an indifferent spell in recent games, showed a flash of his quality by volleying in from 20 yards after a neat lay-off by Constable.
The cushion avoided a tense finale and U's boss Chris Wilder even had the luxury of giving Duberry a brief run-out late on.