Dimitar Berbatov opened the scoring for Sir Alex Ferguson's side at the Rogers Centre, only for Georgios Samaras to equalise for Celtic after a foul by United new boy Chris Smalling.
However, Welbeck and Cleverley both found the target as the Glasgow side ran out of steam after playing on Wednesday when they lost to Philadelphia Union.
Scotland midfielder Darren Fletcher was handed the captaincy, with centre-back Smalling named in the team following his arrival at Old Trafford from Fulham.
United, who had been training in Chicago for four days before the game, made a bright start.
Paul Scholes released Gabriel Obertan in the second minute but his cross was intercepted by goalkeeper Lukas Zaluska.
Then Fabio skipped away down the left and showed good skill to deliver the ball into the area, only for Darren O'Dea to head clear.
United kept up the pressure and Obertan dragged a shot wide in the 10th minute before Berbatov headed over the bar.
Celtic again found themselves on the back foot 19 minutes later when Rafael scampered forward.
He picked out Obertan perfectly but the French winger was wasteful, his cross-cum-shot evading Berbatov.
The Bulgarian striker was further frustrated in the 30th minute when his effort went the wrong side of the post.
However, Berbatov got his reward four minutes later - a cool finish after a superb first touch following a cross from Mame Biram Diouf.
Celtic went close to drawing level in the 43rd minute, only for Charlie Mulgrew's shot to rattle the outside of the stanchion.
Substitute Darron Gibson could have extended United's lead a minute into the second half but blazed his shot over the bar.
Celtic responded and Samaras thought he had equalised in the 49th minute after stepping off the bench. He got in front of goalkeeper Edwin van der Sar and headed firmly into the net but was ruled offside.
Back came United and Diouf was bundled over by Mark Wilson in the 'D' seven minutes later. However, Gibson's free-kick came back off the wall.
Smalling then saw his shot blocked thanks to an instinctive save from Zaluska before conceding a penalty.
The new recruit was penalised for a challenge on Celtic's new recruit Joe Ledley in the 61th minute and Samaras equalised by finding the roof of the net with his spot-kick.
Celtic were growing in confidence and Marc Fortune smashed in a shot after 75 minutes that Van der Sar did well to parry.
However, United regained the lead in the 79th minute with Berbatov setting up Welbeck and the substitute slid the ball home.
It was a good moment for Welbeck who ended the season injured after a loan spell at Preston.
His immediate future remains unclear as Sunderland have already declared an interest in taking him on a similar deal.
Cleverley then added a third four minutes from time in front of a crowd of 39,139 after his 20-yard shot took a deflection off O'Dea and left Zaluska stranded.