The Brazilian's first goal came minutes after Harry Kane's deadlock-breaking header, and he capped Tottenham's biggest win at United in 46 years six minutes from the end.
Spurs join Liverpool, Chelsea and Watford on maximum points from three matches but it is back-to-back defeats in August for United for the first time since the Premier League's debut season.
Seeking a response to their 3-2 defeat to Brighton & Hove Albion, United came flying out of the blocks and had a chance within 20 seconds when Fred latched onto a Jesse Lingard flick-on only to shoot wide.
The Red Devils were very energetic off the ball and soon pressurised Spurs into gifting them a chance when Danny Rose left a back-pass short. Romelu Lukaku rounded Hugo Lloris but sent his shot on the angle wide of an open goal from 12 yards.
The Belgian striker was wasteful again a couple of minutes later when he scuffed Luke Shaw's cutback from 18 yards out. The same combination created United another opportunity but Lukaku could not generate enough power on his header.
Spurs spent much of the opening half-an-hour on the back foot but surged upfield through Lucas on 29 minutes, and the January signing from Paris Saint-Germain should have had a penalty when he was thudded into by Phil Jones.
Referee Craig Pawson gave United a reprieve and the hosts soon regained the initiative to go close through Fred's left-footed curler and a long-distance effort from Paul Pogba which Lloris, playing despite his drink-driving arrest earlier this week, pushed away.
Jose Mourinho's side would have been encouraged by their first-half performance but were hit by a double sucker-punch early in the second period as Spurs struck twice in two minutes after the restart.
Kane climbed above Jones from a 52nd-minute corner to expertly direct a header over David de Gea and underneath the crossbar - his first goal on five trips to Old Trafford.
The hosts were truly rattled when Spurs hit them again moments later, with Christian Eriksen running unchecked into the right channel and pulling a cross back for Lucas to stroke into the bottom corner.
United responded reasonably well but Tottenham stood firm defensively with Toby Alderweireld making several important interventions against the club who were linked with his services for most of the summer.
The Belgian's most crucial contribution was a last-ditch tackle to take a volley off the toe of Lingard 10 yards from goal.
Mourinho's defence missed a player of the same authority and a couple of errors from substitute Victor Lindelof, who replaced the injured Jones, nearly gifted Spurs further goals. The Swede was bailed out by De Gea when he gifted Dele Alli possession on the edge of his own box and then allowed Kane a free header which he missed from six yards.
Fit-again forward Alexis Sanchez also came off the bench and had one half-chance to pull one back for the Red Devils when he drifted in from the left flank to shoot through the legs of Kieran Trippier, but off target.
Lucas landed the killer blow on 84 minutes when he burst past Chris Smalling and buried his shot confidently past De Gea, as Spurs sent a message to Manchester City and Liverpool that they shouldn't be counted out of the title race.
Meanwhile, United are already six points off the pace ahead of next weekend's trip to Burnley.
MAN UTD (5-2-2-1): De Gea; Valencia, Herrera (Sanchez, 55'), Smalling, Jones (Lindelof, 58'), Shaw; Matic (Fellaini, 61'), Fred; Pogba, Lingard; Lukaku
SPURS (4-4-1-1): Lloris; Trippier (Aurier, 76'), Alderweireld, Vertonghen, Rose (Davies, 82'); Dier, Dembele, Alli, Eriksen; Lucas; Kane (Winks, 89')
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