The Ibrox striker broke the deadlock in the 19th minute when he lashed a left-footed shot from 12 yards past Yavuz Ozkan after good work from Steven Naismith and Steven Whittaker down the left-hand side.
However, the Scottish champions were forced to defend for most of the second half and in the 79th minute Sercan Yildirim stabbed the ball past Allan McGregor to give the Turkish club their first Champions League point although they finished bottom of Group C.
Rangers went into the match with no chance of qualifying for the last 16 of the competition but with a place in the Europa League already secured.
However, the Scottish champions were looking for the win which would improve Scotland's UEFA coefficient, net them around 750,000 in prize money and also help their chances of being seeded in UEFA's second-tier club competition.
Walter Smith, however, sprung a surprise by handing 18-year-old Darren Cole his debut as right-back in a five-man defence.
Madjid Bougherra was back in the centre of defence beside David Weir, with Miller again playing the lone striker's role.
However, Bursaspor started the game that little bit sharper than the visitors. They might have taken the lead in the 16th minute when striker Leonel Nunez side-stepped Bougherra in the box to make himself a yard of space, only to fire over the bar.
But three minutes later it was the Scottish champions who grabbed the lead with Miller's first Champions League goal of the season.
Naismith sent Whittaker down the left and when his low cross came into the box, the Scotland striker sent a stunning strike from the penalty spot high past goalkeeper Yavuz Ozkan from 14 yards.
In the 58th minute, Bursaspor skipper Bekir Ozan Has volleyed high over the bar and another opportunity for the Turkish side was missed. However, in the 79th minute Bursaspor got a deserved equaliser when Ozan Ipek's cross from the left found Sercan Yildirim 12 yards from goal and with the Ibrox defence caught flat-footed, he knocked the ball past McGregor. Moments later Yildirim worked a one-two with Insua and found himself through on goal against McGregor - but missed his kick, allowing the Rangers keeper to gather. It was a remarkable let-off for the visitors, who had to defend with some desperation until the final whistle.