Spurs say farewell to their home ground at the end of the season after 118 years and made sure they leave the Lane on a high, Wanyama beating David de Gea with a strong header early on before Kane doubled their lead after the interval with his first goal against United.
Wayne Rooney pulled one back with 20 minutes left, but there would be no comeback as Spurs held on for the win.
Sunday's defeat means United are now out of the race for a top-four finish with two games to go, although Jose Mourinho has been quite vocal in recent weeks that Europa League success is their main priority.
Mourinho will not be overly pleased with how his side's fringe players performed, however, and he could very well decide to shuffle his line-up again when they take on Southampton on Wednesday, with Crystal Palace awaiting next week in their final league game before the clash with the Eredivisie giants.
Spurs got off to a great start and needed just six minutes to open the scoring via Wanyama. Ben Davies curled in a pinpoint cross from the left after Christian Eriksen's corner and the Kenya international rose above his marker before beating goalkeeper De Gea with a strong header.
Anthony Martial came close to levelling in the 18th minute after outmuscling Toby Alderweireld on the edge of the area, but the former Monaco star was unfortunate to see his curled effort go inches wide at the far post.
The home side remained well on top, though, and created a number of chances to double their lead. Son Heung-min fired straight at De Gea after a good run through the middle, Kane saw his header after a corner skim the crossbar, while Dele Alli was denied by the United goalkeeper as well.
But there was no denying Mauricio Pochettino's men after the break and it was Kane who made it 2-0 with a clever finish, poking the ball past De Gea from close range after shrugging off Chris Smalling.
1 - Harry Kane has now scored against 24 of the 25 opponents he has faced in the Premier League (Cardiff being the exception). Portfolio.— OptaJoe (@OptaJoe) May 14, 2017
Mourinho brought on Henrikh Mkhitaryan and Ander Herrera in an attempt to turn things around and United pulled one back in the 71st minute, Rooney finding the net with a clever finish after some good work from Martial down the left.
Alli was then unfortunate not to add a third when Phil Jones headed his shot off the line before Marcus Rashford aimed just wide in the dying moments of the game as it ended 2-1.
Key Opta facts:
- Tottenham recorded their 14th consecutive home win in league competition, equalling their club record previously set between January and October 1987.
- Spurs have gone unbeaten at home for the first time in a league season since 1964-65 (W17 D2 L0 this season).
- Mauricio Pochettino is the first Spurs manager to oversee consecutive home wins over Manchester United in the Premier League.
- The Red Devils suffered back-to-back Premier League defeats for the first time since September 2016 (against Manchester City and Watford).
- In fact, four of Manchester United's five Premier League defeats this season have been in games played on a Sunday immediately after a European match.