Morgan, who had scored a hundred in the opening Test of a troubled summer against Pakistan, was key as England recovered from 47-3 to post 256.
Pakistan started their reply brightly, bringing up the 50 inside eight overs.
But wickets then fell regularly, with Stuart Broad and Graeme Swann taking three apiece, Kamran Akmal top scoring for Pakistan with 41.
Akmal’s dismissal sparked the beginning of the end for the tourists, the innings petering out in his wake as the tourists ended a disastrous tour having lost the Test, Twenty20 and one-day series.
And Pakistan’s collapse finally brought to a finish a series of matches that started in late July.
England triumphed 3-1 in the Test series, 2-0 in the Twenty20s and 3-2 in the one-day matches. On paper, therefore, it has been a summer to savour for the England team, but only the most naive of supporters would expect the results to be what people remember in years to come.
Instead the overriding emotion may well be one of relief – relief that a summer tainted by allegations of spot-fixing against members of the Pakistan team, ridiculed counter-claims made by the Pakistan Cricket Board chairman and occasional flare-ups between players is finally over.
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Yet against the off-field tensions, Pakistan – and Shoaib Akhtar in particular – looked lively from the off, Strauss doubled up by an opening delivery that jagged back sharply into his ribs.
And while there were easy pickings to be had courtesy of an out-of-sorts Abdul Razzaq, England’s innings hit trouble as early as the fifth over.
Steven Davies has had a solid series and surely cemented a place as reserve wicketkeeper for the Ashes tour, but he remains prone to the odd moment of impetuosity.
At 31 without loss and the gentle spin of Mohammad Hafeez on as a surprise first change – he had a decision to make, milk the spinner or attack immediately. He chose the latter, danced down the pitch to a turning delivery that he got nowhere near and was easily stumped by Kamran Akmal.
Apart from Andrew Strauss carting Razzaq over mid-wicket for six, the passage of play that followed was dominated by the visitors.
Jonathan Trott, on to a chorus of boos from Pakistan supporters following his altercation with Wahab Riaz prior to the fourth match, departed shortly afterwards, clean bowled by Shoaib for the third time in succession. Strauss then edged Shoaib behind as England lost two wickets for five runs in six overs.
With the run-rate plummeting as wickets fell, the arrival of a desperately out of form Paul Collingwood was perhaps not the tonic the England fans would have wanted, but the Durham man is nothing if not a fighter.
Initially forced off with a migraine, he returned after Ian Bell had gone for 28. During the time Collingwood presumably spent lying down in a dark room, Morgan had started off another extraordinary innings. Patient at first, brutal at the end and with a couple of new shots thrown in for good measure.
While Morgan went through the gears, seemingly scoring off every ball, Collingwood accumulated. He arrived in the 11th over and departed in the 44th, failing to hit a single boundary in his 47, but the 93 he put on with Morgan helped glacially move the game in England’s favour.
Morgan, though, merely enhanced his reputation as England’s talisman in this form of the game.
He hit glorious cuts and drives among his nine boundaries, managed to work balls pitched well outside off backward of square on the leg side and timed his final assault to perfection, hitting consecutive deliveries for six and four in the final over to bring up his century.
While England finished in a flurry, 19 coming off the final over, the reply started in a similar manner, Kamran Akmal and Hafeez bring up the 50 off only 45 balls. But from there it went downhill, Pakistan playing as if someone had pulled the plug on their innings.
Hafeez, by now completely becalmed, tried a forcing shot off Stuart Broad only to be stunningly caught by a plunging Collingwood, Asad Shafiq following next ball, falling to a routine edge.
The key wicket, though, was that of Kamran. Beautifully set on 41 and looking set for the big innings he had threatened all series he would have been hoping to take the attack to the medium pace of Luke Wright.
Instead, he got a shocking lbw dismissal, given out despite a thumping inside edge, the injustice compounded by the fact Collingwood had been given not out when plumb leg before during England’s innings.
Not that there was any doubt about the next three wickets, Swann bowling Fawad Alam with a beauty that spun from leg to off, Mohammad Yousuf and, first ball, Shahid Afridi.
In the middle of those dismissals, Pakistan brought up their 100, their second 50 having taken 107 balls.
Collingwood, not called upon until the 33rd over, saw his first ball flicked for four by Umar Akmal, but revenge was almost instant, a caught and bowled accounting for the batsman two balls later.
What this shambolic collapse needed was a run-out and two duly arrived.
First Davies reacted sharply to field on the onside and throw to Swann who did the rest to send Umar Gul packing, then Morgan threw down the stumps from cover point with Saeed Ajmal not in the frame.
The coup de grace arrived when Shoaib skewed Broad to that man Morgan. England left to enjoy victory, Pakistan to try to repair damaged reputations.