He levelled the scores with a quarter of an hour remaining after Rangers had let a 1-0 lead slip away.
Dexter Blackstock had put them in front after 32 minutes, but goals from Rowan Vine and Warren Feeney either side of half-time put the visitors ahead.
It took Baidoo to bundle home after a goalmouth scramble to make sure the west Londoners go into the hat for next round's draw.
However, Luton were left furious by the goal, convinced that the youngster had handled.
QPR earlier had strong claims for a penalty turned down on 18 minutes, when Jimmy Smith had his legs taken away from him by Steve Robinson as he went to shoot in the box.
Smith was playing his first game of the New Year, having returned to Chelsea last week to re-sign on-loan at Loftus Road for the rest of the season.
He had the first chance of the game, hitting a sweet 25-yard shot inches over the bar after just four minutes.
Referee Tony Bates showed no interest in his penalty appeals on 18 minutes; but the hosts were dominant and a goal was on the cards.
It came thanks to Blackstock on 32 minutes when he headed home a Lee Cook corner from point-blank range.
Cook had moments earlier forced goalkeeper Marlon Beresford to tip over an almost identical corner, but the Hatters defence did not learn their lesson.
However, this did give them the wake-up call they needed and the visitors were somehow level at half-time.
A neat passing move on the edge of the box was rifled home by Vine on the stroke of half-time - just days after a bid for him from Cardiff City was turned down.
The goal sparked a complete turnaround and the Hatters found themselves in front just a minute after the break.
Ahmet Brkovic's teasing right-wing cross was bravely met by Feeney for only his third goal of the season.
Rangers quickly tried to get back on level terms and Blackstock was unlucky to see his header from another Cook corner blocked on the line by Dean Morgan after 56 minutes.
Boss John Gregory threw on both Baidoo and veteran striker Paul Furlong for the closing stages and the gamble paid off, as the youngster came up trumps and take the tie to a replay at Kenilworth Road.