Christian Burgess gave the Blues the lead early in the second half but Joe Pigott's header and an own goal from Anton Walkes saw the hosts exitÂ the competition in the opening round for the third successive season.
Wimbledon started the better of the two teams and Scott Wagstaff had two early efforts on goal '“Â one of which was hooked off the line by Brandon Haunstrup.
Jackett's side took a while to get going but should have opened the scoring with their first chance in the 23rd minute.
Brett Pitman's cross from the right was met by the rising David Wheeler but the QPR loanee failed to hit the target with the goal at his mercy.
However, Pompey did breakÂ the deadlock four minutes into the second half.
Gareth Evans'Â corner picked out Burgess at the back post and the centre-back kept his cool to powerfully head home from close range.
Neal Ardley's visitors responded well to the setback, though.
Anthony Wordsworth's powerful 20-yard drive on 65 minutes went just over the bar, while Luke McGee made his first real stop of the clash to deny the midfielder with a smart stop six minutes later.
And the Dons got their equaliser in the 76th minute when Pigott was left in space inside the six-yard box and steered Wagstaff's cross past McGee.
WithÂ penalties appearing likely, disaster struck for Pompey with two minutes remaining.
Walkes attempted to clear Tom Soares'Â cross but instead diverted it into his own net.
Pompey have now failed to make the second round of the competition since 2015.
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