United's eighth defeat of a wretched campaign means Liverpool, which currently occupies the fourth and final Champions League place, could go nine points clear of its fierce rival by beating West Bromwich Albion on Sunday.
Charlie Adam scored both of Stoke's goals at Britannia Stadium either side of Robin van Persie's equalizer, with a miserable day for seventh-place United capped by first-half injuries to centre halves Jonny Evans and Phil Jones that forced them off. It was Stoke's maiden win over United in the Premier League, and first since 1984.
"I don't know what we have to do to win," said United manager David Moyes.
Everton bounced back from its derby thrashing by Liverpool in midweek by coming from behind to beat Aston Villa 2-1 and climb to fifth above Tottenham, which drew 1-1 at Hull.
Cardiff climbed off the bottom by scoring two goals in 73 seconds at the start of the second half to defeat Norwich 2-1. That left Fulham, which was overwhelmed 3-0 at home to Southampton, two points adrift in last place.
West Ham completes the bottom three even though it beat Swansea 2-0 thanks to a brace by Kevin Nolan. That win was marred by a red card to striker Andy Carroll for elbowing an opponent.
The other match saw Sunderland earn a second straight 3-0 win at Newcastle in the northeast derby, sparking anger and frustration by home fans disillusioned by the sale of star player Yohan Cabaye to Paris Saint-Germain this week.
Such is the tightness of the bottom half of the table, Sunderland — like Stoke — is now nearer mid-table than the relegation zone.
With Mata joining from Chelsea for a club-record fee last week and both Wayne Rooney and Van Persie back fit, things looked like turning around for United and its under-pressure manager David Moyes.
Most thought the comfortable 2-0 win over Cardiff on Tuesday would be the first in a run of victories to lift the champions up to the top four, but they came unstuck at the hands of a gutsy Stoke side managed by a former United favourite, Mark Hughes.
"I thought we were, to a man, fantastic," said Hughes, who played 467 games for United. "We got a great reward for a huge amount of effort."
Stoke enjoyed some fortune with the first goal, with Adam's long-range free kick in the 38th taking a wicked deflection off Michael Carrick and flying into the far corner past stranded goalkeeper David de Gea.
Mata set up Van Persie to make it 1-1 in the 47th but Adam grabbed what proved to be the winner from 25 yards five minutes later.
"It is a massive result against a great team," Adam said.
Belgium forward Kevin Mirallas popped up with Everton's winning goal from a free kick in the 85th minute, after Steven Naismith had cancelled out Leandro Bacuna's opener for Villa.
Everton is provisionally a point behind neighbour Liverpool, which beat its Merseyside rival 4-0 on Tuesday.
Tottenham moved four points clear of United after Paulinho's 61st-minute goal earned a draw against Hull, which took the lead through Shane Long's first goal for the club since his recent move from West Brom.