Burchill scored the first of three second-half goals as Portsmouth condemned struggling Stoke to their seventh successive defeat.
Stoke had the better of an uninspiring first half, with Portsmouth goalkeeper Shaka Hislop twice denying City striker Tommy Mooney with fine saves.
Hislop needed to be at his brilliant best to keep out Mooney's close-range volley after 20 minutes, with a superb one-handed save after the former Aston Villa and Watford forward had cut in from the left.
Birmingham striker Mooney, starting a three-month loan spell at Stoke, must have thought he had opened the scoring with a brilliant header two minutes before half time.
But the veteran forward was once again denied by a stunning stop from Hislop, who brilliantly punched the ball over the bar as the teams went in 0-0 at half time.
But Pompey did not have to wait long for the opening goal which arrived just four minutes into the second half when the ever impressive Burchill fired home the opener with virtually his first touch of the half.
The 21-year-old Scotsman beat Stoke goalkeeper Neil Cutler to the ball before chipping home from close range after Kevin Harper had crossed from the right.
Pompey had to wait until three minutes from time for their second which arrived courtesy of substitute striker Vincent Pericard.
Pericard, on a season-long loan spell from Juventus, headed home at the near post after Svetoslav Todorov and Portsmouth captain Paul Merson had combined down the right.
Todorov helped himself to the third three minutes into stoppage time, sliding the ball home from ten yards after racing on to Merson's throughball