The second seed and world number three won 6-4 6-3 in an hour and 42 minutes, with the scoreline not representative of a hard-fought match.
Murray was far from clinical, taking only six of the 18 break points that flooded his way, but had enough to see off an opponent who hit 37 unforced errors.
The Scot allowed four break points to slip away in the first game of the match, Cilic finally holding after seven minutes - and it was the Croatian who instead broke at the first attempt to lead 2-0.
Murray hit back immediately and two more breaks followed, leaving the set effectively on serve at 3-2 to Cilic.
That became 4-4 before Murray broke again, served out the set and then broke in the opening game of the second.
The Scot had to battle for more than 15 minutes as he held for a 3-1 lead and the two players traded breaks to reach 5-3, with Cilic serving to stay in the match.
As in the first game some 90 minutes previously, Murray squandered a 0-40 lead and this time wasted three more break - and match - points before finally getting over the line.
Murray told Sky Sports 4: "I think it was a very good match after the first few games, from my side.
"Obviously I had 0-40 in the first game, didn't get it and went a break behind but every time I got broken, I think I broke straight back.
"I came back well from the setbacks and used good variation to keep him off-balance.
"It was a tough match, it was hot and we had some long points, long games, and I'm glad I came through in two sets."
Murray's coach Ivan Lendl had spoken in the build-up to the match about the importance of the British number one reaching number two in the world.
But his player said: "Winning matches is my goal, and by doing that the rankigns take care of themselves.
"I could add extra tournaments to try to get to number two but I want to stick to the schedule I've got and hopefully move up the rankings if I can."
Murray next faces Frenchman Richard Gasquet, who upset fourth seed Tomas Berdych 6-3 6-3, while David Ferrer and Tommy Haas meet in the other half of the draw.