The only goal of the game in the Sanchez Pizjuan came with just eight minutes played, as Freddie Kanoute converted from the penalty spot after Steven Whittaker had fouled Fernando Navarro.
Walter Smith's side were without leading scorer Kris Boyd, who was ruled out due to a thigh strain, while Steven Smith and DaMarcus Beasley were handed rare starts.
The Gers were under pressure right from the off, with Kanoute denied a third-minute opener as Sasa Papac cleared from the goal-line.
However, Kanoute was not to be denied and opened the scoring five minutes later, before Renato tested Allan McGregor with a couple of long-range efforts.
After the restart the home side remained in control without carving out many clear-cut chances.
Ten minutes from time they came close to adding to the margin of victory when Didier Zokora's curling shot came back of an upright with McGregor beaten.