Fraser Aird scored his first goal for Rangers as the Hampden visitors broke the deadlock in injury-time to beat 10-man Queen's Park.
David Templeton shot over the bar and his Rangers team-mate Robbie Crawford headed wide but neither side could land an effort on target in the first half.
Spiders' goalie Neil Parry saved well from Barrie McKay before James Brough was sent off in a scrappy encounter.
Lawrence Shankland twice threatened Rangers before Aird fired the winner.
The league leaders playing away to the team in third place suggested a tight match, but a closer look at the Third Division table gave greater detail to those keen to predict the result.
Rangers had won nine league games in a row since losing to Stirling Albion on 6 October, while the Spiders had won only two of their seven league matches at Hampden this season.
The amateur side had the first chance, though, Ian Watt hitting a free-kick just over the bar on 15 minutes after Willie Collum had penalised Crawford for handball.
Seven minutes later Templeton ought to have broken the deadlock after being picked out in the box by a McKay cross from the left. However, the former Hearts forward was off balance and snatched at his shot, firing over.
There was plenty of bite in the tackles, with Templeton fortunate not to be booked for a foul on David Anderson and Black being shown a yellow card for giving midfielder Paul Gallacher a whack on the ankle.
As the scrappy first half continued, Rangers skipper Lee McCulloch saved the ball from going out on the Queen's Park bye-line and cut it back to Black whose lofted cross was headed a fraction wide by 19-year-old Crawford, making his second league start for the club.
Black lashed a shot a yard wide soon after but the imprecision characterised Ally McCoist's men in front of goal.
Brough was booked for a cynical foul on Templeton in 40 minutes and the Rangers attacker dusted himself down to go on a mazy run through the Queens' defence before being foiled by Gallacher, who read the move well to clear.
Though never a dirty match, the card count mounted in the second half when Kyle Hutton went into the book for leading with his arm while contesting a high ball, then Gallacher was cautioned for a late challenge on Black.
Gardner Speirs' side struggled to threaten Neil Alexander in the Rangers goal but they were organised and energetic in defence to such an extent that it took 54 minutes for the visitors to land a blow on target, Parry throwing up a left hand to save a blistering 16-yard shot by McKay.
But the football merely interrupted the flow of cards.
Brough was upset to have been shown a second yellow when he brought down the flying Templeton with a fractionally late tackle.
Collum then showed a yellow to Anderson for a foul on Black before the Rangers midfielder's substitution for Aird moments later, and took note of Shankland's number when he decided the young winger had dived to win a free-kick.
Watt directed a weak header at Alexander and at the other end Parry came off his line smartly to save at the feet of Lee Wallace after he had worked a one-two with Templeton.
Parry blocked a Templeton shot as Rangers made their one-man advantage count and sub Francisco Sandaza jinked past a few black and white shirts before having his shot diverted by Ricky Little.
The part-timers grew in confidence in the final 10 minutes, surging up the pitch on the break, most notably when Aidan Connolly fed Shankland whose shot, though on target, was too tame.
And, when Wallace made a clumsy challenge just outside the box, it was Shankland who almost snatched the winner with a beautiful curling free-kick that floated a foot wide.
Sandaza hit the side netting with the goal at his mercy and McCulloch headed narrowly over before 17-year-old Aird curled in a shot from 18 yards through a crowded penalty box to secure the points.
Queen's Park: Parry, McGinn, Little, Brough, Robertson, Gallacher, Anderson, Keenan, Watt (Connolly 74), Quinn, Shankland (McParland 89). Subs Not Used: Lochhead, Bradley, Longworth.
Sent Off: Brough (56).
Booked: Brough, Gallacher, Anderson, Shankland, Connolly.
Rangers: Alexander, Hegarty, Perry, Emilson Cribari, Wallace, Crawford, Black (Aird 65), Hutton (Sandaza 71), McKay, McCulloch, Templeton. Subs Not Used: Currie, Cole, Faure.
Booked: Black, Hutton.
Goals: Aird 90.
Referee: William Collum