Despite controlling possession, Louis van Gaal's United failed to muster a single shot on target (for the first time since 2009) and saw their 11-game unbeaten run in all competitions ended when Tadic netted in the 69th minute.
The Serbian's goal gave Southampton a first win at Old Trafford since January 1988 and enabled them to avenge their 2-1 loss in December's reverse fixture, when United had managed only three attempts at goal.
It took Ronald Koeman's side two points above their opponents into third place, while United, who have won only once in their last five league games, now lead fifth-place Arsenal by a single point.
United had new goalkeeper Victor Valdes on the bench and welcomed Daley Blind back from injury, while Angel di Maria made his first start since November, but Radamel Falcao was absent from the match-day 18.
United had not played at home since Boxing Day and as they took time to reacquaint themselves with home comforts, Southampton almost capitalised, with the fit-again Nathaniel Clyne shooting wide in the first minute.
Di Maria, who had been sidelined by hamstring and pelvis problems, started up front alongside Robin van Persie, but although he flashed a few shots wide, United struggled to bring him into the game.
Southampton, who have now won four times in five league outings, flooded the midfield, frustrating United captain Wayne Rooney in his attempts to create openings from deep.
The visitors' plans were compromised in the 19th minute when Toby Alderweireld was forced off after appearing to hurt his thigh as he cleared Rooney's pass towards Di Maria.
But Florin Gardos, who replaced the Belgium international, produced an important block to thwart van Persie before half-time.
Southampton forced United onto the back foot in the early stages of the second half and it took smart blocks from Chris Smalling and Saints old boy Luke Shaw to deny Steven Davis and Pelle in turn.
United manager van Gaal took action by introducing Ander Herrera and Tyler Blackett in place of van Persie and Shaw, which saw Rooney move further forward and Blind take up a place at left wing-back.
But it was Southampton substitute Tadic -- sent on in place of Southampton debutant Eljero Elia -- who made the most telling contribution.
Five minutes after coming on, the Serbian released Pelle and when the Italian's low shot came back off the foot of the right-hand post, Tadic, following up, stroked the rebound home.
Van Gaal sent on the calvary, throwing on Marouane Fellaini for Di Maria, but still United failed to test Fraser Forster in the visitors' goal.
Juan Mata was guilty of two glaring misses, shooting wide of an unguarded goal after Forster had swatted away a cross from Blind and then clipping over from six yards when the Dutchman bent another centre into the box.
Blind also volleyed wide from an Antonio Valencia cross, but despite referee Phil Dowd awarding five minutes of stoppage time, United could not avert a fourth defeat of the campaign.
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