A number of Premier League scouts were reported to be in attendance to watch the highly-rated Wickham, and the 17-year-old did not disappoint, completing his treble from the penalty spot after Rovers defender James O'Connor handled in the final minute.
It capped a miserable night for Doncaster, who were largely the architects of their own downfall.
Sam Hird headed into his own net after 25 minutes and another misjudged aerial clearance from the same player allowed Lee Martin to tee up Colin Healy for his first Ipswich goal.
England Under-21 star Wickham netted either side of the interval and Gareth McAuley marked his 100th league appearance for Town with a goal.
Paul Jewell has now guided Ipswich to four wins in six outings since taking the reins at Portman Road. By contrast, Doncaster have now lost six in eight, rendering their new year play-off dreams a distant memory.
Top scorer Billy Sharp missed out for Rovers after sustaining heavily bruised ribs in the weekend defeat to Portsmouth.
Brian Stock was also an injury victim and Matthew Kilgallon dropped to the bench, meaning Joseph Mills, Mark Wilson and Dean Shiels came into Sean O'Driscoll's starting XI.
Jason Scotland replaced Tamas Priskin in Ipswich's attack and Healy started in place of captain David Norris, who failed to recover from the knee and ankle knocks he suffered in the 1-1 draw at Barnsley.
In the 11th minute Martin twisted and turned on the Doncaster right before Wickham prodded his cross goalwards. Gary Woods saved well in the Rovers goal and he was grateful to see the subsequent ricochet off defender James Chambers land in his arms.
Doncaster's first shot on goal came 23 minutes in when John Oster surged forward from midfield before driving narrowly past the post.
The visitors hit back immediately and Woods looked on as Jimmy Bullard's deflected shot from the corner of the box spun wide after Wickham had brought down Carlos Edwards' cross.
A spell of set-piece pressure followed and Ipswich took the lead in bizarre fashion.
Edwards' shot was blocked and diverted skywards, leaving Hird to challenge for the ball with McAuley. The Rovers defender jumped highest, but his header spiralled agonisingly into his own net.
Hird's hapless evening continued when he failed to deal with a long ball, leaving Martin time to pick out Healy who slotted in from 12 yards to make it 2-0 after 33 minutes.
Another Doncaster mistake effectively ended the contest three minutes before half-time.
Oster carelessly conceded possession in his own half, allowing Healy ample space to line up a rasping 25-yard drive. Woods parried but Wickham was on hand to gleefully thump home.
Coppinger tried to haul his side back into the match, first capitalising on a slack pass from Edwards to test Fulop from distance and then crossing for Mills to head over in first-half stoppage time.
Ipswich's fourth came after an hour when Woods could only punch Bullard's left-wing corner as far as Wickham, who emphatically volleyed into the centre of the goal.
The trickle of home fans leaving the Keepmoat Stadium became a procession when McAuley's header from Bullard's searching free-kick squirmed past Woods into the bottom corner via the upright.
Damien Delaney and Healy both went close as Bullard's set pieces continued to mystify the Doncaster backline.
The hosts' best attacking moment came in the closing stages when Fulop produced an excellent double save to deny O'Connor and Paul Keegan, before Wickham had the final word.