The Scot looked assured in the opening stages of the match and Dodig gifted Murray a break of serve early in the first set to hand him the advantage.
Despite Murray's serve being uncharacteristically inconsistent, the Olympic champion's ground strokes were too much for Dodig and the third seed triumphed 6-2 in the opener.
Murray looked in full flow in the second set, with the British number one capitalising on some sloppy play from Dodig to claim three breaks of serve.
Dodig's serve looked weak in the closing stages and Murray extended his lead with a 6-1 triumph.
The Arthur Ashe Stadium were treated to more quality tennis by Murray in the third set, with the Wimbledon finalist dictating play easily from all areas of the court.
Dodig showed some fight despite an inevitable loss edging closer for the Coratian but Murray's range of shots proved too much for his opponent as he found the crucial break he needed to wrap up the match in straight sets.