Wimbledon looked to be cruising through after Adebayo Azeez and Lyle Taylor had put the hosts two goals ahead after just 25 minutes.
But Jake Jervis pegged them back just before the break before Ryan Brunt and McHugh completed the comeback in the second half.
The game was only seven minutes old when incisive play from Azeez released Andy Barcham, who had two shots at goal before the former slotted home easily to hand the home side an early lead.
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Wimbledon then took advantage of Argyle's lack of set-piece finishing and extended their lead through Taylor after Azeez and Tom Beere combined well.
The visitors got on the scoresheet after 38 minutes through Jervis after he took on a long ball and confidently placed it under James Shea.
After Azeez fired over when one-on-one Gary Sawyer crossed for Brunt to head home a 54th-minute equaliser.
And from their next attack McHugh punished the hosts for failing to clear a cross as he fired Plymouth into the next round.
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