The highly-rated 18-year-old striker, whose performances for the Tractor Boys have attracted admiring glances from a clutch of leading Premier League clubs, struck in the 24th minute of Saturday's Championship clash after David Norris' deadlock-breaker at Turf Moor.
Norris swept home a 17th-minute opener with a neat left-foot finish and Wickham's strike put Paul Jewell's men 2-0 up and in total control.
Jay Rodriguez hit back for Burnley in the 68th minute with a well-taken strike but there was to be no equaliser for Eddie Howe's men as Ipswich held on.
The Clarets' hopes of a top-six finish are beginning to fade and they have now picked up just one point from their last four matches.
Ipswich are effectively playing for pride in the remainder of the campaign as they are safe from the drop but too far adrift of the top six to realistically claim a top-six berth.
Burnley began the brighter as winger Ross Wallace drilled a shot wide from 18 yards in the eight minute.
Yet the visitors gradually steadied themselves and they began to move through the gears.
In the 16th minute, Tractor Boys striker Jason Scotland engineered the space to hit a shot from 12 yards which Tyrone Mears did superbly well to block.
The respite did not last long for the hosts, however, as an outswinging corner from Jimmy Bullard fell to Norris and he fired home left-footed from 12 yards to put Ipswich 1-0 up.
They should have claimed a second moments later after Bullard's free-kick fell to Scotland six yards from goal but he fired over when well placed.
The bustling Scotland then saw a shot blocked by Shane Duffy, but his side's dominance was soon rewarded when a delightful move culminated in Wickham finding the net.
A fine passing sequence saw the ball worked out to Norris on the right flank and he crossed for Wickham to head in from 12 yards out.
Clarets boss Eddie Howe made a tactical switch in the 37th minute, replacing Wallace with targetman Chris Iwelumo.
Chris Eagles then fired over the bar in the 44th minute and midfielder Dean Marney then forced Arran Lee-Barrett into a smart stop.
Eagles went close again and on-loan Aston Villa striker Nathan Delfouneso also threatened before Rodriguez gave the Clarets hope with 22 minutes remaining.
The England Under-21 striker collected a pass from Marney and drove a right-footed shot into the net from 20 yards.
With 15 minutes remaining, Howe replaced Marvin Bartley with Graham Alexander but the visitors held out for all three points.