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.
The visitors finished the tie with 10 men, as Nathan Thompson was dismissed for a second bookable offence.