Both sides went into the contest searching for their first victories but it was Mark Hughes’ Potters who came through with a comfortable win against David Moyes’ beleaguered Black Cats, thanks to Allen's first-half brace.
The result moves Stoke out of the relegation places on six points while Sunderland stay rooted to the bottom on two points.
Sunderland had plenty of the early possession but went behind in the eighth minute.
Marko Arnautovic got played in towards the by-line and his pull-back picked out Allen in the six-yard box, and he headed in his third goal in as many Premier League games from close range.
The Black Cats went close to an equaliser in the 17th minute when Wahbi Khazri sent a lovely cross into the area, but Ryan Shawcross put in a vital block to deny Jermain Defoe a certain goal.
Defoe had another opportunity on 26 minutes when played into the area by Khazri, but this time his effort flashed just over Lee Grant’s crossbar.
Just before the half-hour mark Wilfried Bony, looking for his first goal on the year, lashed a low strike the wrong side of the Sunderland upright.
On 36 minutes Arnautovic was close to making it 2-0 to the hosts, but the Austrian hitman toe-poked his effort wide from a Bony return pass in the area.
However, Stoke did get the second of the game right on half-time when a corner into the area was not dealt with by the Black Cats and the ball fell to Allen, who rifled in through a packed area for his second of the game.
Jordan Pickford kept Sunderland in the game early in the second half when he did well to deny Bony in the area before Papy Djilbodji blocked Xherdan Shaqiri’s follow-up effort.
Two minutes later Bony again went close but fired his early strike wide of the post inside the area.
Charlie Adam stepped off the bench on 65 minutes and within three minutes he teed up Arnautovic to score from close range, but the effort didn’t count as the offside flag was up.
Adam almost capped the win with a third goal on 88 minutes but his effort from distance crashed off the crossbar and bounced clear.