But Mick McCarthy's side were forced to work hard against the spirited effort of the struggling Cheshire side, who are without a win in 11 games now.
Yet they made life difficult for the Wearsiders, who had to weather a late storm in which David Vaughan came close to bagging an equaliser but for Steve Caldwell's vital block.
Colin Murdock saw Marcus Stewart clear his firm header from a corner early on and Mark Rivers then flashed a volley a couple of yards over.
But Sunderland hit back and George McCartney's 25-yard effort forced Ben Williams to make a double save.
Stewart then wasted a good opening when Michael Bridges dummied Dean Whitehead's crossfield pass and Stewart spun away, but fired high and wide from the edge of the box.
It almost proved costly for McCarthy's men as Steve Foster thumped a downward header inches wide of Thomas Myhre's far post from Vaughan's carefully-flighted free kick just past the half-hour mark.
But Crewe also had an escape when attacking left-back McCartney whipped in a cross from the left on 35 minutes. The ball sneaked through to the far post to hand Carl Robinson a sight at the bottom corner, but under pressure from Anthony Tonkin the Welsh international failed to get in a shot and Williams held on to the half-hit shot.
McCarthy attempted to pep up his disappointing frontline nine minutes after the break replacing Michael Bridges with Elliott and the Irishman was swiftly into the action turning away from Foster and driving just over the top corner.
But the introduction of a second substitute in the space of three minutes proved decisive with Julio Arca taking the ball almost as soon as he had replaced Sean Thornton and steaming clear of Darren Moss.
The Argentinian's shot from the left of goal came back off the post and Elliott picked up the rebound and smashed it home from eight yards.
Rivers was inches away from applying a finishing touch after Luke Varney flicked on a Kenny Lunt corner.
Then Vaughan wanted too much time when Varney turned on Lee Bell's excellent pass and set the winger up on the left of the box. But Whitehead was in quickly with some mean defending to take the ball out for a corner.
Foster snuffed out Elliott's chance of a second after Stewart pulled the ball back from the by-line after some neat play by Arca as Sunderland looked to kill the contest.
But they were forced to endure some uncomfortable moments in the closing stages. Jeff Whitley's late foul on Vaughan offered Lunt a free-kick within his shooting range in the last minute, but the Alex captain fired it over.