Rangers more than matched them in the early stages yet Hearts still landed the first serious blow.
In the 10th-minute some super skill from Deividas Cesnauskis took out two defenders and freed Paul Hartley, who squared the ball for Edgaras Jankauskas to side-foot home from close range.
Although top scorer Rudi Skacel quickly forced an important save from Ronald Waterreus the visitors held the edge in a congested midfield area and Peter Lovenkrands fired inches wide following a rare slip by Julien Brellier.
Dazzling wing play from Chris Burke created a terrific chance for Rangers shortly before half-time, but there was nobody on hand to convert his tempting cross.
Hearts were sharper after the break as the game really opened up, but in the 55th minute Scotland keeper Craig Gordon had to pull off a wonder save from Thomas Buffel after a free-kick was deflected into his path.
Ten minutes later Buffel finally got the better of Gordon when he arrived at the back post to squeeze another inch-perfect Burke cross into the net with his left foot.
Rangers sensed the opportunity to grab all three points and striker Dado Prso twice went close and also had a penalty claim refused.
Burke shot wide when he should have done better and the blonde winger then picked out Buffel yet again for another fizzing effort which flashed wide.
The closing 10 minutes belonged to Hearts and full-back Robbie Neilson almost won it for them with a spectacular 30-yard half-volley which smashed back off the bar.