Peter Lovenkrands continued his rich vein of scoring form as Rangers made up ground on second-placed Hearts.
The Danish striker pounced after 35 minutes, stabbing the ball home from Hamed Namouchi's headed knock-down.
Chris Burke and Lovenkrands twice went close and Marvin Andrews missed a golden close-range chance as Rangers enjoyed long spells of superiority.
Hearts' hopes faded when Saulius Mikoliunas was sent off for a bad challenge on Barry Ferguson.
The Ibrox side remain fourth and are now 11 points behind the Tynecastle team, who have won just one of five games under head coach Graham Rix.
Neither goalkeeper had much to do in a scrappy opening 20 minutes, with both sides working hard to close each other down and lots of passes going astray.
Rangers winger Burke looked lively in the early stages and his scampering runs were causing the visitors some concern.
The home side came close when Namouchi met a Bob Malcolm free-kick and bulleted a header on target but Craig Gordon pulled off a fine save to tip the ball just over the bar.
Steven Pressley wasted a great chance when Ibrox keeper Ronald Waterreus elected not to go for a cross and the Hearts captain's header from close range was just too high.
Burke gave the Hearts defence a hard time at Ibrox
Andrews then produced an incredible miss, when he scooped a shot straight up in the air from almost underneath the crossbar.
Barry Ferguson released Burke into the box only for his point-blank header to be blocked by the onrushing Gordon.
Rangers' pressure paid off when a Malcolm free-kick was knocked into the path of Lovenkrands by Namouchi and the striker found the net from inside the six-yard box.
Rangers could have doubled their advantage shortly after the restart but Lovenkrands fired over the crossbar after a neat cut-back from Burke.
Lovenkrands went close with a header and Burke drilled a shot inches wide as Rangers continued to press forward.
With six minutes remaining, substitute Mikoliunas was given his marching orders for upending Ferguson.
Rangers: Waterreus, Murray, Andrews, Kyrgiakos, Burke, Ferguson, Malcolm, Namouchi (Thompson 79), Buffel, Smith, Lovenkrands (McCormack 90). Subs Not Used: Klos, Novo, Pierre-Fanfan, Jeffers, N'Guessan.
Booked: Malcolm. Goal: Lovenkrands 35.
Hearts: Gordon, Neilson, Pressley, Webster, McAllister, Cesnauskis (Mikoliunas 64), Hartley, Brellier, Skacel, Jankauskas (Elliot 64), Bednar. Subs Not Used: Banks, Simmons, Berra, Wallace, Tall.
Sent Off: Mikoliunas (86). Booked: Bednar, Pressley.
Ref: M McCurry