Keane has endured a difficult relationship with Town supporters following a dire run of form which saw them slip to 16th in the Championship following Sunday's 4-1 defeat to East Anglia rivals Norwich.
However, Keane's battling team - hit by injuries and unavailable loan players - secured their place in the last four through a second-half penalty from Grant Leadbitter.
West Brom made nine changes for the trip to a freezing Suffolk, with youngster James Hurst, 18, making his debut at right-back, as defender Marek Cech and midfielder Graham Dorrans were the only survivors from the 4-1 Premier League win at Everton.
Keane had accepted in his programme notes "there is no hiding place for a manager", and the side he put out certainly did not shy away from their challenge.
The 10,000 or so home supporters who had turned out almost something to cheer inside the first two minutes when Hungarian Tamas Priskin got away down the right and his low cross into the six-yard box cannoned off captain David Norris and just over the angle.
West Brom looked dangerous on the counter-attack, with Marc-Antoine Fortune's angled strike pushed behind by stand-in keeper Brian Murphy.
It should have been 1-0 on 15 minutes when Priskin brushed off Pablo Ibanez as they chased into the right side of the West Brom box, before cutting the ball back to Jason Scotland - but his close-range effort was straight at Boaz Myhill.
At the other end, Giles Barnes latched onto a high ball into the Ipswich box to stab an angled shot goalwards, which Murphy blocked.
Moments later and the Ipswich keeper was backpedalling to touch Dorrans' cross over the bar and then flapped at another ball into the box, which Simon Cox could only hook over.
The Town faithful appealed for a penalty when the ball ricocheted up onto Steven Reid's arm, but referee Michael Jones waved play on before Leadbitter then tested Myhill with a low drive, which rebounded just away from Tommy Smith.
Ipswich were awarded a free-kick in a central position, some 22 yards out, when Cox barged over Blues captain Norris.
Leadbitter chipped the pass back up neatly, but his volley was straight at Myhill.
Town were again screaming for a penalty when Priskin went down under Myhill's diving challenge, but once more neither the referee or his nearside assistant were interested - much to Keane's annoyance.
Ipswich made a change at the start of the second half when Reggie Lambe was replaced by Carlos Edwards.
West Brom had a chance when the ball was floated into the penalty area and Cox turned to guide it just wide of the far post.
Ipswich took the lead from a controversial penalty on 69 minutes.
Edwards stole possession off Marek Cech and scampered away down the right.
He charged towards the area, and was tripped by Dorrans. While contact looked just outside the box, the referee pointed to the penalty spot.
Leadbitter made no mistake from 12 yards to put the Championship side within reach of the semi-finals.
West Brom brought on Ishmael Miller, for Fortune, as Roberto di Matteo looked to breathe fresh life into his attack.
Ipswich pressed for a second to kill off the tie - and almost had it when Scotland's close-range strike hit Priskin on the line.
Cox saw his shot on the turn deflected over and Murphy saved well from Ibanez at point-blank range.
Priskin broke clear, but then drilled wide - and after three minutes of stoppage time as the snow fell on Portman Road, it was the Blues who took their place in the last four.