The result means that Burnley have failed to win any of three home games under Coyle despite a successful series of away performances.
Burnley's uncertainty on their home patch was evident early on as David Unsworth was caught in possession by Rowan Vine, but his pull-back was well defended at the expense of a corner by Graham Alexander.
Alan Mahon wasted a rare opportunity, taking a pass from Kyle Lafferty on the edge of the box before shooting well wide with a left-foot effort.
Vine seized on more hesitation at the heart of the home defence, but under pressure from Clarke Carlisle he fired over the crossbar.
As the first half drifted towards its conclusion, Mahon struck a shot wide but then went close from 20 yards after taking a pass from Robbie Blake and striking a shot against the post.
Burnley suffered a blow early in the second period when central defender Carlisle limped off and was replaced by Joey Gudjonsson.
But it was Rangers who were making the clear-cut opportunities with Scott Sinclair playing a one-two with Vine only to see his shot well saved by Gabor Kiraly.
Vine and Sinclair combined again moments later but Kiraly blocked at the expense of a corner.
Burnley's relief was short-lived however as Sinclair's delivery to the far post was headed home by an unmarked Damion Stewart.
In response, Coyle replaced Blake and Mahon with Steve Jones and James O'Connor and the changes seemed to bring new life to the home side.
Lafferty had a deflected shot well saved by Lee Camp at the expense of a corner and the keeper produced an even better save to deny substitute O'Connor a possible equaliser.
Both Gudjonsson and winger Wade Elliott found the side-netting with pressure mounting on the R's defence.
However, it was the visitors who were to spring a surprise by sealing the game in stoppage time with a bizarre goal as Burnley keeper Kiraly went up-field for a corner only for Rangers to break out of defence and secure the points as Vine stroked the ball into an empty net.