The 67-year-old Frenchman won't discuss his future until after Saturday's FA Cup final against Chelsea.
The win kept the Manchester side two points ahead of Liverpool, whose comfortable success at home to Middlesbrough earned them a place in the Champions League playoff round.
Firmino was only prevented from adding a third by a George Friend block shortly afterwards, but the hosts would not be denied for much longer and Lallana put the match - and a top-four finish - beyond any doubt in the 56th minute when he provided a composed finish at the end of a quick counter-attack.
The contract renewals of Mesut Ozil and Alexis Sanchez became major issues and while a switch to three at the back helped spark an upturn in form with Arsenal winning their final five Premier League games, it was not enough to clamber back into the Champions League places this time.
If Liverpool beat relegated Middlesbrough at Anfield and City avoid defeat at struggling Watford, then, even if Arsenal win, Wenger's side will be consigned to the indignity of Thursday nights in the unglamourous Europa League next term.
Kane fired home in the 11th minute and netted again two minutes later. Chelsea won the last meeting, 3-1, back on February 4.
Joy at Anfield, anger at the Emirates - Liverpool do the necessary to qualify for the Champions League, while Arsenal and Wenger must acquaint themselves with the unfamiliar environment of the Europa League.
Blues boss Antonio Conte paid a fitting tribute to Terry, taking him off in the 26th minute - matching his shirt number - and Chelsea's players formed a guard of honour as he walked off to a standing ovation by the capacity crowd.
Despite United's lowest finish since Alex Ferguson retired after collecting the Premier League trophy in 2013, the League Cup has already been collected and Sunday's game offered hope for the future. Chelsea's 30th win of the campaign was sealed by Eden Hazard, Pedro Rodriguez and a double from Michy Batshuayi.
With United due to face Ajax in the Europa League final on Wednesday, Mourinho made no attempt to hide his desire to keep his key players fresh for the Stockholm showpiece.
The Gunners have finished the last 20 years in the top-four but face the possibility of dropping out of the race for a Champions League spot today. The northwest side lost 2-1 to West Ham but finished 16th, six points clear of danger.