Chelsea increased their advantage over Manchester City despite Diego Costa's 11th minute close-range header being answered eight minutes later by Dusan Tadic's fortuitous penalty in an absorbing encounter at Stamford Bridge.
Yet United, so often criticised for being slow and directionless under Dutchman Van Gaal, produced the most electric performance of the day, crushing Spurs with first-half goals from Marouane Fellaini, Michael Carrick and Wayne Rooney.
A national newspaper ran a bizarre story on Sunday about Rooney being filmed getting knocked out by his friend, Stoke City player Phil Bardsley, in a boxing bout in Rooney's kitchen.
Rooney's amused response after his goal was to celebrate by throwing a few shadow punches before falling theatrically on to his back amid delighted cheers from the United faithful.
Chelsea, on 64 points, still have a game in hand on Manchester City, who were beaten at Burnley on Saturday.
United, with 56 points, have climbed within a point of Arsenal and two of their cross-city rivals City as well as moving five points clear of fifth-placed Liverpool.
Their win also seriously dents the Champions League qualifying hopes of Spurs who, on 50 points, are now behind sixth-placed Southampton on goal difference and struggling to keep alive their hopes of a top-four finish.
TRAUMATIC WEEK
At Stamford Bridge, a traumatic week for Chelsea, knocked out of the Champions League by Paris St Germain at the last 16 stage and heavily criticised for the on-pitch behaviour of their players, ended with a little more frustration.
Manager Jose Mourinho, though, made light of not picking up the three points. "I agree Chelsea not winning at home is not a good result, but we had a five-point lead before the match and afterwards we have a six-point lead," he said.
"Now we have six points more and one fewer match to play. It makes our situation better than before the game. It's positive rather than negative."
Southampton caused problems in the opening period but Chelsea asserted their authority after the break with Costa clipping the post while captain John Terry was also denied in a frantic finish.
Mourinho was also unhappy that, after Costa capitalised on inexplicably static defending from Southampton to head home Branislav Ivanovic's cross, Southampton were awarded a penalty when Sadio Mane was adjudged to have been felled by Nemanja Matic.
Tadic converted from the spot, although his tame effort was fortunate to beat the trailing feet of Thibaut Courtois.
SUBLIME UNITED
United, reeling from Monday's FA Cup defeat by Arsenal, produced a performance Old Trafford had been longing for as they blew fellow top-four chasers Spurs away with the pace, precision and directness missing so often in Van Gaal's unloved reign.
Fellaini, released by Carrick's intelligent through ball, put them ahead after nine minutes with a sweet left-foot strike, before Carrick headed home from distance 10 minutes later after the excellent Fellaini had wrought more havoc.
Rooney's bustling run and cool finish after 34 minutes provided the knockout blow.
In Sunday's other game, Everton eased pressure on manager Roberto Martinez with their first home league success in three months with a 3-0 win over 10-m1n Newcastle United, with goals from James McCarthy, a Romelu Lukaku penalty and Ross Barkley.
Newcastle had Fabricio Coloccini sent off just before the hour for a foul on Aaron Lennon.
It was Everton's second win in their last 13 Premier League matches.
