The youngster, who celebrated his 21st birthday on Saturday, suffered a nasty clash of heads with team-mate Graeme Lee in the third minute.
As Moncur and Lee lay motionless on the ground in the penalty area the ball ran loose and Grant Holt fired
his shot against Rhys Evans' legs and Ben Davies coolly netted the rebound.
Bradford challenged referee Jarmil Singh's decision to allow the goal as staff from both sides tended the injured defenders.
Both played on after lengthy treament, but Moncur was rushed to hospital when he collapsed 10 minutes later though subsequent scans cleared the player who was released.
City, who were under the cosh for most of the first half, enjoyed the best of the second period but could not fashion an equaliser.
Their second successive defeat was confirmed in the third minute of stoppage time when Holt broke away and centred for sub Richard Walker to score from close range.
McCall said: "We knew they would attack us down the wings, but when Moncur was injured we completely lost our shape.
"We knew we were going to a tough place and Shrewsbury got in seven or eight crosses in the first half when we needed Moncur there.
"In big games like this the first goal is vital. They got it and we got disrupted. But I have no excuses," he added.
Shrewsbury: Daniels; Hurd, Jackson, Coughlan, Tierney; Davies, Thornton, Cansdell-Sherriff (Ashton 80), McIntyre; Holt, Hibbert, (Walker 67). Subs not used: Garner, Langmead, Humphrey.
Bradford City: Evans; Moncur, (Nix 23), Clarke, Lee, Heckingbottom; Colbeck, Furman, Bullock, (M Boulding 57), McLaren, Daley; Conlon.
Subs not used: Bower, McLaughlin, R Boulding.
Referee: J Singh (Middlesex).
The full article contains 314 words and appears in n/a newspaper.