Leeds skipper Struijk scored the first league goal conceded by the Blades in seven matches and substitute Joseph added a late second.
Daniel Farke's side leap-frogged the visitors into second place and extended their unbeaten league run to five matches.
Sheffield United boss Chris Wilder has quickly reinvigorated the South Yorkshire club following their miserable season in the Premier League and they looked on course to frustrate their hosts and eke out a vital point.
But Struijk's goal gave Leeds a deserved lead on the balance of play at Elland Road on Friday night, while Joseph's effort was only the fifth league goal the visitors have conceded this campaign.
Tributes were paid before kick-off to former Blades defender George Baldock, who died at his home in Greece last week aged 31.
A minute's silence was spontaneously turned into a minute's applause by both home and away fans, while the players, including Socceroos defender Harry Souttar, wore black armbands.
Sheffield United players stand in tribute to former Blades right-back George Baldock. (AP PHOTO)
Leeds began to find their rhythm midway through the first half and Largie Ramazani twice threatened to give the home side what would have been a deserved lead.
The forward struck the outside of a post from a tight angle and then forced Blades goalkeeper Michael Cooper into a brilliant low save.
American Brenden Aaronson and Willy Gnonto went close as Leeds finished the first half on top.
Scoring chances remained a rarity and Aaronson blazed another effort over the crossbar before Junior Firpo mis-cued in the box.
The Blades briefly threatened after Joe Rodon's mis-placed pass but the Leeds defender recovered to block Harrison Burrows' shot and Ollie Arblaster's stinging drive deflected off Ao Tanaka for a corner.
The hosts then broke the deadlock, becoming the first side to score a league goal against United since August 24.
Rothwell swung over a corner from the left and Struijk was first to the ball, sweeping a first-time left-footed shot high into the net.
But Leeds were not yet done and in the final minute Piroe's clever pass picked out Joseph, a 79th-minute replacement for Ramazani, and the striker out-muscled the Blades defence to fire home.
