The 21-year-old Irishman, on loan from Sunderland, was stretchered off in the 12th minute with a suspected broken right leg after falling awkwardly after heading the ball – the game was held up for five minutes while he received attention.
Captain Gary Jones scored the crucial goal in the 22nd minute after right back Stephen Darby hoisted the ball into the goalmouth. Leading scorer Nahki Wells slipped as he tried to shoot and, when the ball ran loose to the unmarked Jones, he scored with a 10-yard shot.
Plymouth, now without a victory in seven games, almost pulled themselves level in the six minutes the referee had added on at the end of the first half following Egan’s injury when Nick Chadwick headed the ball down for the unmarked Jamie Lowry.
But Bantams goalkeeper Matt Duke made a superb save to turn away the midfield player’s fiercely-struck shot.
The visitors had another great chance to score five minutes after the interval when Chadwick flicked the ball on to Alex MacDonald, who outpaced defender Rory McArdle, but then sent a rash shot wide of the far post.
Duke was forced to come to the home side’s rescue again when he turned Chadwick’s goal-bound shot from an acute angle over the crossbar in the 78th minute.
But Phil Parkinson’s team held on for a win that keeps them in the play-off positions.
Bradford City: Duke, Darby, McArdle, Egan (McHugh 17mins), Meredith, Atkinson, Doyle, G Jones, Wells (Connell 77mins), Thompson (R Jones 80mins), Hanson. Unused substitutes: McLaughlin, Naylor, Forsyth, Ravenhill.
Plymouth Argyle: Gilmartin, Berry, Blanchard, Nelson, Bhasera, Lowry, Hourihane, Cowan-Hall (Harvey 74mins), Chadwick, MacDonald (Feeney 90mins), Gurrieri. Unused substitutes: Cole (gk), Feeney, Madjo, S Griffiths, Lennox, Richards.
Referee: T Bates (Stoke).