The Clarets’ dynamic duo Danny Ings and Sam Vokes had fired the visitors into a second-half lead which would have taken them above Rangers and up to second on goal difference.
But West Ham flop Maiga, who had moved to Loftus Road on loan barely 12 hours earlier having scored just one goal in 15 Premier League appearances this term, climbed off the bench to net a precious equaliser.
Another debutant, Kevin Doyle, had headed Rangers into the lead and Richard Dunne fired them ahead again after Ings’ equaliser.
But Vokes’ second-half double put Burnley in charge until Maiga’s late strike.
While Rangers’ promotion hopes are being hindered by a lack of goals, Ings and Vokes now have 37 between them – one more than Harry Redknapp’s side have managed in total this season.
Shot-shy Rangers’ cause has not been helped by a season-ending shoulder injury to their 15-goal top scorer Charlie Austin, which forced Redknapp into a familiar deadline-day scramble for new recruits.
One of them, Wolves loanee Doyle, took just six minutes to make impact when he powered in a header from Niko Kranjcar’s corner.
Burnley quickly regrouped, though, and hauled themselves level in the 25th minute when Kieran Trippier’s cross was volleyed home on the turn from the edge of the area by Ings for his 22nd goal of the season.
Ings’ first-time finish was impressive but it was topped nine minutes later by the unlikely figure of Dunne.
Another Kranjcar corner was headed out to the right corner of the area, from where Dunne expertly volleyed it straight past Tom Heaton and celebrated his first goal since January 2012.
But Burnley hit back again nine minutes after the interval when Vokes got ahead of Dunne and prodded Michael Kightly’s cross in with his studs.
And eight minutes later Ings forced Benoit Assou-Ekotto into a costly slip on the byline and rolled the ball across goal for Vokes to tap into an empty net.
Redknapp promptly threw Maiga, whose first touch of note was a shot which, familiarly to West Ham fans, flew about six yards wide.
But the Mali international was on hand to tap in a cross from Junior Hoilett with 12 minutes left to snatch a point.