Murray, who reached the French Open semifinals last week despite injuring his ankle earlier in the tournament, was leading 3-1 against the 2002 Wimbledon semifinalist when play was interrupted by a heavy downpour.
The second-seeded Briton wrapped up the first set when the players returned, but Malisse cut out the errors in the second and snatched it when he broke for the first time at 6-5 after Murray went long with a forehand.
But Murray broke early in the decider and again in the last game to set up a third-round meeting with Janko Tipsarevic of Serbia.
Short link: