The defender slammed home in the 83rd minute to add to Issa Diop's earlier strike while Javier Hernandez grabbed a third in stoppage time.
This was effectively a training match for the Hammers as the League One Dons played for 72 minutes with ten men after Rod McDonald was dismissed early on.
It was a huge blow for Neal Ardley's side who had taken a shock lead in the second minute.
Pellegrini had looked to be taking no chances and named a strong line-up for the Carabao Cup tie.
After disappointing defeats to Liverpool, Bournemouth and Arsenal, the Hammers needed a shot in the arm of confidence ahead of this weekend's Premier League tie with Wolves.
Pellegrini made nine changes from the side that lost 3-1 to the Gunners and handed full debuts to Andriy Yarmolenko, who cost £17.5million from Borussia Dortmund, and Spaniard Lucas Perez.
He also included £22million defender Diop while club record signing Felipe Anderson, who cost £36million from Lazio in the summer, came on at half time.
While West Ham's summer spending topped the £100million mark, Dons Ardley spent around £100,000 on brining 10 new faces to the club.
The majority of which was on striker James Hanson from Sheffield United, while Ardley also included home-grown defender Toby Sibbick and midfielder Anthony Hartigan in his XI.
But it was a £50,000 duo plucked from non-League who combined to shock the Hammers and put the Dons in front inside two minutes.
Mitchell Pinnock, signed from Dover for around £25,000, floated a corner into the box and Joe Pigott powered his header past Adrian and into the roof of the net.
It was the former Maidstone man's fourth goal of the season as Pellegrini's side got off to the worst possible start.
They eventually found their feet as Yarmolenko and Pedro Obiang had shots from distance that went over the bar.
And they were soon given another boost as the Dons were reduced to ten-men in the 18th minute when Rod McDonald was shown a second yellow card.
The centre-back, who picked up his first booking for a reckless challenge on Robert Snodgrass, was sent off for hauling down Javier Hernandez just five minutes later.
McDonald's dismissal saw Pigott replaced by Will Nightingale, as Wimbledon plugged the gap in their defence with another player from their academy.
And with the extra man, West Ham started to press forward for an equaliser as Declan Rice and Snodgrass went close.
Dons captain Deji Oshilaja then cleared Yarmolenko's looping header off the line while Nightingale also made a number of crucial tackles to ensure the Dons led at the break.
Pellegrini was worried and hauled off Obiang for Felipe Anderson at the break and the £36million man from Lazio first contribution was an overhit pass that went through to Dons keeper Tom King.
West Ham continued to attack in waves but came up against a resilient Dons defence as Oshilaja, Nightingale and King continued to frustrated the Premier League side.
Hernandez saw his header saved by King before Anderson fired wide before the Mexican international missed a sitter.
Yarmolenko powered his way down the right and cut the ball back for Hernandez, who dragged his shot wide from just six yards.
An equaliser was on the cards and it came on 63 minutes as Diop eased the pressure on Pellegrini with a thumping effort from outside the box.
The £22million defender, who scored an own goal in the defeat to Arsenal, lashed home and gave King no chance.
The giant French Under-21 international almost had a second two minutes later when he connected with Anderson's corner only to see his header smack the crossbar.
Hernandez scuffed another shot wide and Yarmolenko squandered another as West Ham set up camp inside the Dons' half.
West Ham were relentless but so too were plucky Wimbledon, who defended for their lives.
But they were breached again in the 83rd minute when Ogbonna swept home from Snodgrass's cross.
And with the Dons defence flagging, Hernandez finally got his goal as he latched on to Arthur Masuaku's pass before rounding King.
It was tough on the Dons, who had defended for the majority of this match and were hoping for penalties.
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