They are, indisputably, England's team of the decade and, if they win the FA Cup at Wembley next Saturday, and the European Champions' Cup in Barcelona on Wednesday week, will have a decent claim to be the best of the century.
Cole, having surprisingly been left on the bench, scored a beautifully taken goal, his 24th of the season, one minute and 47 seconds after coming on as a half-time substitute. His strike earned United a lead they never relinquished, David Beckham having cancelled out Les Ferdinand's shock 24th-minute opener two minutes before the break.
The victory meant Arsenal's 1-0 win over Aston Villa at Highbury, secured by a goal from another substitute, Nwankwo Kanu, was in vain. Twelve months after winning the Double, Arsenal, who have contributed so much to this season, end it with nothing. Football can be a capricious lover.
The game's romance this year is with Manchester United and, despite their many previous affairs, there was nothing blase about the way they celebrated the consummation of this one. Peter Schmeichel, who marked his last match at Old Trafford with his fifth championship medal, was as jubilant as Dwight Yorke, who was winning his first. Alex Ferguson, their 57-year-old manager, celebrated like a teenager.
He told the adoring masses: "What a season, what a fantastic bunch of lads, they've been absolutely magnificent." He added: "We played with the quality that is expected of Manchester United and also the spirit which is the important thing."
Schmeichel then bade farewell, after eight years at the club. To the backdrop of fans shouting "Don't go" he said: "To finish like this is really fantastic. Me and my family have really enjoyed our stay in Manchester."
The vibrant celebrations culminated an afternoon on which the Theatre of Dreams, which has often been so quiet it appeared the crowd really were slumbering, rocked with noise. Beforehand Russell Watson, a local tenor, had raised the roof with a ringing rendition of "Nessun Dorma" and the fans had responded in kind.
The selection of Teddy Sheringham, ahead of Cole, may have raised some doubts, along with the absence of Jaap Stam, but the prevailing mood was one of confident anticipation. When David Ginola limped off, after an innocuous-looking tackle from Gary Neville, that sentiment hardened into certainty.
That was hardly surprising. Already Ian Walker had made a sharp save from Dwight Yorke at the near post then seen the ball bounce back into his arms off the post after one of his clearances had hit the Tobagan.
Heaven knows what the reaction would have been at Highbury if that goal had gone in but any doubts they had about Tottenham's commitment soon evaporated. Steffen Iversen had already volleyed over when he rose above David May to flick on Walker's long kick and Ferdinand, getting ahead of Ronnie Johnsen, chipped cleverly over Schmeichel.
Watched by the assistant manager and chief scout of Bayern Munich, their European Cup final opponents, United rolled into the attack. After 32 minutes Walker made a brilliant double save from Scholes, then Stephen Carr blocked a goalbound shot from Yorke while Beckham and May headed over.
Finally Scholes dispossessed Tim Sherwood, exchanged passes with Ryan Giggs then, showing great awareness, picked out Beckham arriving late into the box. Walker got a hand to the shot but its ferocious power carried it past him and in off the post. "Are you watching Arsenal", sang the Spurs fans.
The interval arrived and so did Cole. Almost immediately he was stealing on to Gary Neville's pass and looping the ball over Walker. The Tottenham 'keeper then saved twice from Scholes, who put another chance wide as did Nicky Butt. With Yorke and Cole also failing to score a clinching goal , nerves crept in.
"Their central midfield had started to get hold of the ball and it was a hanging-on job," Ferguson admitted. "We were hanging on to a whole season of hard work."
One man remained nerveless, Roy Keane, magnificent despite his ankle injury. In the quiet moments, as he jogged back into midfield after some lung-bursting run, he could be seen with a limp but, whenever the ball was near, he was making himself available for a pass or rushing to close down an opponent. In the closing minutes, when a rare Tottenham attack induced panic in supporters and players alike, it was Keane who tidied up with a neat interception, and Keane who then thumped the ball clear. When, a few minutes later, the title was won it was Keane who Ferguson went to first.
"How fit was he? He was 150 per cent because he wanted to be 150 per cent fit," said Ferguson of his captain. He added: "We deserved the title because we are the best team in the country. We have to take ourselves to the wire all the time but we proved there is something about certain teams that elevates them beyond the rest. We just never give in."
Goals: Ferdinand (24) 0-1; Beckham (43) 1-1; Cole (47) 2-1.
Manchester United (4-4-2): Schmeichel; G Neville, May, Johnsen, Irwin; Beckham, Keane, Scholes (Butt, 70), Giggs (P Neville, 79); Sheringham (Cole, h-t), Yorke. Substitutes not used: van der Gouw, Solskjaer.
Tottenham Hotspur (4-4-2): Walker; Carr, Scales (Young, 70), Campbell, Edinburgh; Anderton, Sherwood, Freund, Ginola (Dominguez, 9; Sinton, 77); Ferdinand, Iversen. Substitutes not used: Clemence, Baardsen (gk).
Referee: G Poll (Tring).
Booked: Manchester United: Sheringham. Tottenham Hotspur: Anderton.
Man of the match: Keane.
PREMIERSHIP HONOURS LIST
Champions: Manchester United (into Champions' League)
Runners-up: Arsenal (into Champions' League)
Third place: Chelsea (into Champions' League qualifying competition)
Uefa Cup: Leeds (fourth in Premiership), Tottenham (Worthington Cup winners), Newcastle (FA Cup finalists); possibility of further place via Uefa Fair Play League (would go to highest team in Premiership Fair Play League)
Intertoto Cup: West Ham (four semi-finalists in Intertoto Cup, which is staged in the summer, go into Uefa Cup)
Relegated: Charlton, Blackburn and Nottingham Forest
Promoted (from First Division): Sunderland, Bradford City and play-off winners (Bolton, Ipswich, Watford or Birmingham)