An own goal from Leeds goalkeeper Illan Meslier in the 14th minute had handed the Blades a halftime lead, and after Junior Firpo headed the visitors level, Tanaka and Piroe struck in the final three minutes to secure victory.
Both managers, Chris Wilder and Daniel Farke, had earlier played down the notion this clash was a title decider with 12 games still to play, but Leeds have moved five points clear of second-placed Blades and, crucially, seven above Burnley in third.
Farke was serving a touchline ban after receiving his third yellow card of the season last week for celebrating his side's stoppage-time winner against Sunderland.
The German looked on from the television gantry as Sheffield United set about his side from kick-off, with Leeds' bid to ride the early storm undermined by Meslier's jitters.
The Frenchman had been in fine form since being at fault for all three goals in a 3-3 draw at Hull in January, but looked all at sea on Monday (Tuesday AEDT).
Callum O'Hare's header from Ben Brereton Diaz's teasing cross was tipped on to the crossbar and down on to the goal-line by Meslier. Tyrese Campbell headed the rebound against a post, but the Leeds goalkeeper failed to gather the ball and instead parried it into his own net.
Leeds steadily wrestled back control and began to create chances of their own, Firpo hauling the visitors level with a thumping 72nd-minute header.
The visitors finished with a flourish, Tanaka heading into the top corner following Firpo's header with two minutes left. Then in the 90th minute Piroe lashed home Leeds' third to seal victory.
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