Wednesday keeper Kevin Pressman had kept his side in the match with a series of stunning saves before Owusu struck in the 73rd minute with Shefki Kuqi adding the second seven minutes from time.
The 34-year-old keeper foiled John-Paul McGovern, Iffy Onuora and Stuart McCall with eye-catching stops before Owusu broke the deadlock.
Wednesday could have snatched the lead in the opening minutes when Craig Armstrong's corner was flicked on to Kuqi who produced a tame header that was easily collected by Paddy Kenny.
United teenager Michael Tonge weaved his way into the Wednesday area minutes later and pulled a low shot across the face of the goal as McGovern tried to connect at the far post.
Kuqi squandered another half-chance in the 22nd minute when he connected with another Armstrong corner but this time he headed over the bar from ten yards out.
Pressman then foiled McGovern with a stunning fingertip save as he pushed a stinging shot round the post after Wednesday failed to cut out Carl Asaba's low cross.
McGovern's corner was then met by striker Onuora eight yards out as Pressman flung himself to the right to tip the ball towards the far post as defender Ian Hendon chested it down and cleared off the line
Wednesday, outplayed for most of the game, almost found a way through when Kenny failed to hold a Kuqi header but Rob Ullathorne was alert enough to clear the ball off the line. Minutes later Paul McLaren saw his overhead kick crack against the bar.
Pressman pushed away a 25-yard strike from McCall before Owusu struck in the 73rd minute. Kuqi saw his headed smash against the bar and the Wednesday substitute cracked in the loose ball from close range.
Kuqi headed home the second from Alan Quinn's cross with seven minutes to go and there was still time for Asaba and substitute Peter Ndlovu to throw away gilt-edged chances in the final minutes as United looked unsuccessfully for a way back.
Sheff Wed: Pressman, Hendon, Bromby, Burrows, Westwood, McLaren, Soltvedt, Armstrong, Donnelly, Kuqi, Knight. Subs: Evans, Quinn, Owusu, Hamshaw, Maddix.
Sheff Utd: Kenny, Kozluk, Ullathorne, Yates, Murphy, McCall, Jagielka, Tonge, McGovern, Asaba, Onuora. Subs: Brown, Ten Heuvel, Montgomery, Doane, Ndlovu.