Ex-Don Ali Al-Hamadi gave the Blues a third-minute lead, but the League Two hosts hit back with goals from Omar Bugiel and Matty Stevens either side of the break to get Plough Lane rocking.
The party was in danger of going flat when Conor Chaplin levelled for the Tractor Boys on 86 minutes, but the Dons held their nerve to triumph 4-2 on penalties and earn a juicy third round tie at home against Newcastle.
Isaac Ogundere netted the winning spot-kick after Dons keeper Owen Goodman had saved from Jack Taylor and Omari Hutchinson.
"We said to the lads there will be no repercussions if it (the penalty shoot-out) doesn't go our way, so hopefully that brought them a bit of calmness," a delighted Jackson told BBC Radio London. "They took them really well.
"It's the highlight of my career, certainly one of them.
That's the third time coming up against Kieran (McKenna) as a manager and it hasn't gone great for me in the previous couple, so to win tonight against such a talented young coach is nice on a personal level.
"But I'm just really pleased for the football club to have a night like this. Beating a Premier League team is outstanding."
Fellow League Two side Barrow also took the scalp of higher level opposition, knocking out Championship newcomers Derby 3-2 on spot-kicks after a goalless draw. It took the Bluebirds through to the third round for the first time since 1967-68 and they can now look forward to a glamour trip to Chelsea.
After James Collins and Kayden Jackson missed their penalties for Derby, Barrow keeper Paul Farman saved from Craig Forsyth to seal the upset.
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END OF CUP ROAD FOR EFL BATTLERS
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