The home side wasted a host of chances as they had the better of a mediocre game at Portman Road but looked set for an 18th draw of the season as both sides were reduced to 10 men in the second half.
But substitute Wickham popped up to score a fine solo effort to give Keane's men a bit of breathing space at the bottom.
Both sides remain in danger of relegation to League One and the visitors will have to do without captain Cliff Byrne for a lengthy suspension after he was dismissed for the third time this season.
He was later joined by Ipswich defender Damien Delaney as both sides finished with 10 men.
Captain Jon Walters had the first shot on goal in the second minute, but saw his effort clear the crossbar from 25 yards before Daryl Murphy went close for the hosts.
Pablo Counago was recalled to the starting line-up and after he broke free down the left he picked out Murphy lurking inside the area but the on-loan Sunderland man dragged his shot wide.
Scunthorpe striker Paul Hayes was played in down the left and he found Matt Sparrow running into the box. Sparrow lashed a shot goalwards with his left foot but saw goalkeeper Brian Murphy make an excellent reflex save to turn the ball away for a corner.
Keane's side continued to press and hit the crossbar in the 41st minute.
An Owen Garvan corner was cleared back out wide, and when Garvan whipped in a second cross Counago rose highest to flick a header over goalkeeper Joe Murphy but onto the face of the crossbar.
Hayes was denied by Brian Murphy before the break before Daryl Murphy hobbled off with an injury to be replaced by Wickham.
Garvan then crossed for Walters to loop a header toward the top corner but Joe Murphy back-pedalled to tip over, with Gareth McAuley heading wide from the following corner.
Counago then nipped in after some defensive hesitation to round Joe Murphy but he took to long to get his shot away and Byrne was able to block on the line.
It was fine work from Byrne but he then blotted his copybook in the 69th minute when he was shown a straight red card for body checking Jack Colback as he looked to run onto a through ball behind the defence.
The decision looked harsh and it wasn't long before Delaney was also dismissed by referee Pat Miller.
Substitute Jonathan Forte knocked the ball past the former Hull defender and after he stepped across to bring the striker down Miller showed him a straight red card.
The game looked set for a stalemate but Wickham had other ideas.
A long goal-kick was flicked on to him and after holding off two defenders the teenager steered a shot wide of Joe Murphy and into the corner.