The Scot, who had been a little off form in his opening match against Alex Bogomolov, appeared to be back to his best as he broke the Dodig serve six times in total on his way to a 6-2 6-1 6-3 victory.
The British number one, who won singles gold at London 2012, will face 30th seed Feliciano Lopez or his fellow Spaniard Pablo Andujar in the third round, with the match slated for Saturday.
"It was better than the first round, that's for sure," Murray said of his performance. "It was pretty comfortable conditions out there today. There wasn't really any wind, it was fairly cool as well. That helped. I moved better than I did in the first match and served better, and I was able to dictate more of the points because of that. I was much happier with the way I played."
There was further good news for British tennis when Laura Robson produced a coming-of-age performance to defeat ex-world number one Kim Clijsters in the women's singles.
Robson, 18, saw off the Belgian 7-6 (7/4) 7-6 (7/5) in Arthur Ashe Stadium to send the four-time grand slam winner into retirement. Clijsters had long earmarked the US Open as her last tournament before quitting.
"I've always loved playing on a big stage," said Robson, who now faces China's Li Na. "I think I thrive in a good atmosphere and I feel like I play better on big courts when I have a lot of support. It's easier when you have less pressure on you."