The world No 3 teamed up with fellow Scot Colin Fleming to beat Ivan Dodig and Mate Pavic in the doubles, the British pair winning 6-3, 6-2, 7-6 (8-6), 6-1.
There had been concerns the Wimbledon champion could miss yesterday's rubber because of a back injury but, apart from the occasional grimace, Murray looked in fine form.
He said: "Colin played exceptionally well, bar a couple of points in the third set, and we dominated the majority of it."
Dodig awaits Murray in singles action today and the Scot believes he has the measure of the world No 32 after their doubles meeting.
"You get a feel for the return and I returned his serve fairly well today," Murray said. "I felt I got a pretty good read of it which will hopefully help tomorrow."
Fleming could only appreciate the quality of his partner. "Andy's returns are phenomenal," he said. "He's a great player. He's able to figure things out during the match, knows when momentum is swinging, and it's great to have him in the team."
Victory for Murray over Dodig in the first reverse singles rubber today would see Leon Smith's team return to the elite World Group for the first time since 2008.
Murray and Fleming broke the Croatian pair as early as the second game, with the help of a double fault from Pavic and that was all the advantage they needed to take the first set.
Another break followed in game three of the second set thanks to a winning forehand from Murray. An error from Dodig in the seventh game secured a double-break and Fleming then served out for a two-set lead.
Pavic was broken again in the opening game of the third set but Croatia broke back immediately. Fleming secured a further break but that was wiped out once again. Britain opened up a lead in the tie-break but Croatia held off a match point and went on to take the set after a double fault from a nervous Fleming.
However, any fears over the British pair, and Fleming's confidence in particular, were quickly brushed aside as they went on to take the fourth set comfortably with the help of some woeful serving from Pavic.