Mick Harford's men must have fancied their chances of bagging all three points when Bantams full-back Paul Heckingbottom was sent off for a second bookable offence 18 minutes from time.
But a defensive mix-up gifted City the opener with just 11 minutes left and it took a powerful header from Spillane four minutes from time to grab a point.
With injuries taking their toll on Luton's resources, Harford started the game with just Chris Martin up front, but his hard-working side snuffed out City's dangermen for long periods and the Hatters boss was left feeling disappointed that they had not won the game.
Luton's injury problems increased as both Sol Davis and Bradford old-boy Lewis Emanuel both picked up first-half knocks and there were few clear-cut chances in a scrappy opening period.
City keeper Rhys Evans did well to reach a long-range strike from Martin as the interval approached and his opposite number, Conrad Logan, bravely smothered the ball at the feet of Bantams striker Michael Boulding a minute later.
Promotion-chasing Bradford, who had lost their two previous matches, upped the pace after the break and Logan did well again to tip over a thumping shot from Paul McLaren after a well-worked free-kick.
City winger Joe Colbeck had a sight of goal just after the hour mark, but the Hatters defence shut him down as he lined up a strike from just inside the box.
Luton centre-half Harry Worley was lucky to get away with a handball appeal from the City players, but there could be no argument about Heckingbottom's dismissal after he clattered into Martin.
City boss Stuart McCall threw on substitute Barry Conlon to beef up his attack and he was rewarded soon after when Spillane and his team-mates failed to deal with a long ball into the box.
The defender miscued his attempted clearance and it fell kindly into the path of Conlon, who slid the ball into an empty net from ten yards.
But Harford's young side showed plenty of spirit to claw their way back and Spillane made amends in full when he raced onto a well-delivered cross from Ed Asafu-Adjaye and headed home the equaliser.