Rangers 0 Hibernian 1
The only goal came just after the hour mark when David Murphy beat keeper Allan McGregor to Filipe Morais' fiercely struck corner to glance his header into the net.
Murphy was also the Hibs' hero at the other end when he headed a David Weir effort off the line after keeper Yves Makaba-Makalamby was caught out of position.
The away side's flowing football had Rangers chasing shadows for much of the afternoon.
The home side started brightly and Barry Ferguson's overhead shot from Kris Boyd's header went narrowly over.
Hibs were also moving the ball about well, but Mickael Antoine-Curier went too early and drifted offside.
Brahim Hemdani chested the ball to Ferguson, whose turn and shot was well blocked by the Hibs keeper.
Just on the stroke of half-time, Rangers went close to taking the lead.
Alan Hutton's pass released Steven Whittaker whose cross was palmed out by Makalamby to Charlie Adam, but from eight yards his lob went narrowly wide.
Early in the second half after Hutton was fouled on the edge of the area, Boyd's fiercely-struck shot was well saved by the keeper diving to his right.
Steven Fletcher glanced his header narrowly over before Hibs took the lead.
McGregor looked slow to move towards Morais' corner and Murphy leapt in front of him to head home.
Hibs nearly added to the lead but Fletcher shot wide, before Rangers made a double substitution with Lee McCulloch and Steven Naismith coming on for the last 20 minutes.
The home side stepped up the pressure but Boyd's strong appeals for a penalty after his flick appeared to strike Rob Jones on the arm were turned away.
From an Adam free-kick, Carlos Cuellar headed back across goal but from only three yards Weir shot over.
Still Rangers continued to press and in a last throw of the dice Daniel Cousin came on for Boyd.
When Makaba-Makalamby was again caught out of his goal, Sasa Papac lobbed the ball back in for Weir to head goalwards only for man-of-the-match Murphy to head clear from under his bar.