View the photos from Pompey's cup loss at Luton
James Collins netted the all-important goal in first half stoppage-time after being teed-up by Danny Hylton.
But Pompey had plenty of chances to secure at least a replay, with Marek Stech making a superb save to keep out Matt Clarke's header.
And the Hatters keeper also had to be at his best to prevent Ben Close's effort from finding the bottom corner at the death.
The visitors finished the tie with 10 men, as Nathan Thompson was dismissed for a second bookable offence.