The Bramall Lane side had cantered in to a 3-0 lead with goals from Henri Camara and Jamie Ward before struggling Plymouth hit back.
United started in explosive fashion, Richard Cresswell heading narrowly wide before Camara snatched the lead on seven minutes.
A towering header from centre-back Chris Morgan was flicked on to Mark Yeates who threaded the ball to the ex-Wigan Athletic striker. Camara shrugged off Kari Arnason's tepid challenge before tucking the ball past the advancing David Stockdale and into the far corner.
Ward added the second on 35 minutes. The goal was set up by full-back Tony Kallio's deep cross from the left. Camara failed to connect as the little winger finished clinically, lashing the ball high into the net from 15 yards for his seventh of the season.
United's third came in the 47th minute, Nyron Nosworthy's long throw falling nicely for Ward who turned quickly to drill past Stockdale for his seventh of the season.
Subsitute Yannick Bolasie made it 3-1 in the 48th minute, stabbing the ball home from close range after keeper Mark Bunn failed to hang on to Arnason's long throw.
Joe Mason added a second for the Pilgrims in the 57th minute before Richard Cresswell appeared to kill off the visitors when he made it 4-2 ten minutes from time.
However, Jamie Mackie made it 4-3 in the 87th minute with a spectacular 30-yard shot and Mason almost grabbed an equaliser seconds from time, his low shot flashing inches wide of the target.