The Scot required just one hour and 20 minutes to wrap up a 6-3 6-2 success over the world number 68.
Murray made his breakthrough in the fourth game of the opening set when he broke Stakhovsky's service game to open up a 3-1 advantage.
From that point Murray went on to serve out the set 6-3.
Two further breaks of Stakhovsky's serve followed in the second set as Murray advanced through to round three of the competition.
Murray is the second seed at the event behind Rafael Nadal.