A run of four defeats in five league games had sapped City's confidence, but Marc Bridge-Wilkinson's third minute strike revived their self-belief and they ran out comfortable winners thanks to three second-half goals.
New loan signing Colin Healy impressed on his City debut and the home side's dominance would have earned greater reward in the first period but for some fine saves by Crewe keeper Stuart Tomlinson.
Dario Gradi's injury-ravaged line-up had barely settled when Bridge-Wilkinson made his mark as City counter attacked from a Crewe free-kick.
A vital header from Bantams skipper David Wetherall snuffed out the danger and when the play was quickly switched upfield, midfielder Steven Schumacher fed the ball through to Bridge-Wilkinson who suddenly found himself racing into the Crewe box.
Tomlinson sprinted out to close him down, but the City man's shot ballooned towards the goal and despite the efforts of two defenders, the looping ball found its way into the back of the net.
City have rarely taken the lead in their recent games and with their tails up they pegged back Crewe looking for a second.
The visitors had a let off when striker Dean Windass raced clear onto a long punt from his keeper. His lofted shot beat Tomlison, but on-loan defender Danny O'Donnell got back to head the ball clear.
Tomlinson then made a fine save to push away a low drive from flying winger Jermaine Johnson and the Crewe custodian was in the right place to deny Windass and former Alex man Eddie Johnson in quick succession.
Stripped of their hot-shots Nicky Maynard and Luke Varney, the visitors failed to force a save out of City keeper Donovan Ricketts in the first period and it was thanks to Tomlinson that Windass didn't add to City's opener.
After 32 minutes the experienced striker forced his way into the Crewe box but Tomlinson got down smartly to parry his low right foot shot.
Six minutes before the interval Crewe almost snatched an unlikely equaliser when Luke Rodgers set up Rodney Jack on the edge of the Bradford box but the winger's shot was blazed high and wide.
Tomlinson rode his luck just after the break when he spilled a Bridge-Wilkinson corner only to see the ball bounce off Johnson back into his grasp and five minutes later Bradford wasted a great chance to land another sucker punch.
Another counter-attack stretched the Crewe defence to the limit, but Windass' pass to Johnson was over-hit and the chance went begging.
Soon after Tomlinson excelled again saving a Schumacher volley from point-blank range, but the Crewe keeper could do nothing when the hard-running City man added their second after 61 minutes.
Bridge-Wilkinson supplied a teasing cross from the left and Schumacher could hardly miss as he raced onto the inviting ball.
The goal knocked the stuffing out of Crewe and City took full advantage to book a safe passage into the second round with goals from the hard-working Windass and luckless Crewe substitute Darren Kempson.
Windass finally got his reward when Johnson's neat pass sent him surging into the Crewe box and the City hot-shot gleefully blasted a 12-yard strike beyond Tomlinson after 70 minutes.
Bradford were now fully in charge and it was no surprise when they bagged a fourth.
Substitute Ben Muirhead again tested the Crewe back line and although Kempson stopped the ball reaching the waiting Windass, his intervention could only divert it past the stranded Tomlinson.