Thiem took the upper hand in his match against former world number three Cilic when he broke in the ninth game on his way to sealing the first set.
Cilic, who made 39 unforced errors, was flustered by Thiem’s defence, suffering two more breaks in the second set.
Two poor volleys in the sixth game of the third earned the two-time runner-up the crucial break en route to victory.
The Austrian third seed will play American qualifier Jack Sock in the next round.
Thiem, 27, said he was happy he managed a successful transition from playing on hard courts in the heat of New York two weeks ago to performing on clay in damp 13C conditions in Paris.
“I’m from Austria so know how it is to play in cold conditions,” he added. “When I was in the juniors I played in 10 degree weather.”