Bryan Robson's promotion favourites were no match for the Home Park side, who could have had more goals but for keeper Ian Bennett and the woodwork.
Plymouth were handed a half chance in the opening minutes as Lee Martin burst into the area, but the midfielder seemed to scuff his shot as Bennett scooped up the ball.
United were caught out on 15 minutes as Sylvan Ebanks-Blake chased a long ball downfield pursued by Matt Kilgallon before the centre-back stretched out a foot to snuff out the danger as the little striker was about to shoot.
Danny Webber showed his pace minutes later darting past his marker as he cut inside before unleashing a shot from 20 yards which drifted wide as Romain Larrieu scrambled across his line.
Plymouth snatched the lead in the 24th minute as the Sheffield defence was caught napping.
Ebanks-Blake threaded the ball to Halmosi who shrugged off Gary Cahill's challenge to finish with a low strike beyond the exposed Bennett as he raced off his line.
Nick Montgomery squandered a chance minutes later after Webber unselfishly picked him out but, with Larrieu off his line, the midfielder scooped the ball wide of the target from 10 yards.
United, who were totally stifled in midfield, were almost caught out on the stroke of half-time.
Ebanks-Blake's angled drive from 16 yards was pushed away by Bennett before Halmosi hammered in a shot from the edge of the area which crashed against the woodwork.
The United keeper kept out a long-range strike from Halmosi before leading scorer James Beattie at last sparked into life with a desperate shot from almost 30 yards which ballooned high over the bar.
The impressive Ebanks-Blake almost grabbed a second with spectacular strike from over 30 yards but the ball curled beyond the keeper before hitting the bar.
Plymouth were foiled twice by Bennett as he blocked a stinging Ebanks-Blake effort, before ending Martin's daring run into the area as the Home Park side threatened to over-run the Blades.