Spurs plan to have their new stadium, built on the same site, ready for the 2018-19 campaign and will play their home matches at Wembley next season.
Mauricio Pochettino’s side left their current ground, where they have spent 118 years, on a high by staying unbeaten there this season and securing second spot in the Premier League.
Wanyama got Spurs off to the best possible start with a header five minutes in and Kane doubled their lead early in the second half, flicking home from a Christian Eriksen free-kick.
Captain Wayne Rooney gave United hope of a recovery when he poked in from Anthony Martial’s low cross, but they were unable to spoil the leaving party.
Defeat means Jose Mourinho’s men cannot now finish in the top four.
They can still qualify for the Champions League if they win the Europa League, but Mourinho will have to settle for fifth or sixth place in his first season at the club.
Farewell to the Lane
Tottenham’s title chances ended last week with their defeat by West Ham, and the trophy went to Chelsea on Friday.
With a Champions League place already guaranteed, Pochettino said their final home match was all about making it a special day for the fans.
The teams walked out to a display of flags around the ground, Spurs legends including Chris Waddle and Glenn Hoddle were invited as special guests, and local musical duo Chas and Dave provided half-time entertainment.
And the Tottenham players gave supporters a first-half performance to remember.
Ben Davies’ sublime cross was headed home by Wanyama to give Spurs the early advantage, and they could have extended their lead before the break had goalkeeper David de Gea not denied Son Heung-min and Kane.
Five minutes into the second half, the hosts doubled their lead. Eriksen’s free-kick curled into the path of Kane and he out-smarted defender Chris Smalling to poke home his first goal against United.
Spurs’ performance dropped off after that, but they managed to hold on – despite Rooney’s goal giving them a scare.