The Dane - who was making his first appearance in 18 days - netted after just 11 seconds, before Phil Jones put through his own net later in the first half to give Spurs a deserved three points from the encounter in North London, which was played in front of a Premier League record crowd of 81,978.
United under-performed as they lost their six-game run of clean sheets, but both teams received a boost with the news that Chelsea had lost at home to Bournemouth.
All eyes were on former Arsenal forward Alexis Sanchez when the game kicked off at the national stadium but attention soon switched to Eriksen, who was celebrating a goal after 10.4 seconds having capitalised on a Harry Kane flick and Dele Alli's deflected pass before slotting his finish into the far corner after 10.4 seconds.
United almost netted an immediate equaliser through Jesse Lingard, with Hugo Lloris coming out well to make the block, and the game continued at a ferocious pace without many clear-cut chances being created.
However, Spurs were slowly getting on top and moments after Alli had volleyed wide from 14 yards, a second goal arrived via the boot of United defender Jones, who diverted Kieran Trippier's cross inside David de Gea's near post from eight yards out.
Soon after the goal, Anthony Martial forced a smart save out of Lloris from an acute angle but as the first half approached a conclusion, it was Spurs who looked most likely to add another.
Eric Dier headed over the crossbar after meeting an Eriksen free kick, while Kane sent an effort off target after receiving the ball behind the United defence.
Spurs had the better of the play after the restart, but United came close to halving the deficit when Romelu Lukaku held off the attentions of Jan Vertonghen before drilling a half-volley towards the top corner, only to see Lloris tip his strike over the crossbar.
Just after the hour mark, Mourinho introduced Marouane Fellaini for Paul Pogba but the alteration only led to Spurs looking more likely to add a third, with Eriksen firing marginally wide of the post from distance and Son Heung-min seeing a strike from an acute angle deflected wide by De Gea.
Within 10 minutes, Mourinho decided to withdraw Fellaini but despite the suggestion that the Belgian midfielder was being brought off for tactical reasons, the United manager revealed after the match that the midfielder has sustained a knee injury.
The change did not help the visitors as Spurs continued to create opportunities at will, with Kane twice missing the target and Dier seeing a curling effort deflected wide of the post.
Kane continued to search for his 100th Premier League goal during the closing stages but while the forward would endure a frustrating night, his side were able to move to within five points of second-placed United in the standings.
TOTTENHAM HOTSPUR (4-2-3-1): Lloris; Trippier, Sanchez, Vertonghen, Davies; Dier, Dembele (Wanyama 91'); Eriksen, Alli (Sissoko 88'), Son (Lamela 80'); Kane
MANCHESTER UNITED (4-2-3-1): De Gea; Valencia, Smalling, Jones, Young; Matic, Pogba (Fellaini 63', Herrera 70'); Sanchez, Lingard (Mata 63'), Martial; Lukaku