Cilic ranked 12th, got out slowly in the first set, but controlled the final two en route to the second round. He won 89 percent of the points when he converted his first serve, but managed only six aces to Dodig's 13 in the match that lasted 1:46.
Cilic was the only seeded player on the men's side of the tournament to play on Monday. Third-seeded John Isner and Sam Querrey, the tournament's No. 4 seed, are expected to open play Tuesday.
On the women's side, top-seeded Kirsten Flipkens of Belgium defeated Shahar Peer of Israel 6-2, 6-2, while second-seeded Sofia Arvidsson of Sweden topped Slovakia's Jana Cepelova 6-4, 6-4.
In a late match Monday night, No. 8 Kristina Mladenovic of France took just less than an hour to defeat wild-card entry Courtney Collins of the United States 6-3, 6-1.
Cilic and Dodig grew up in the same neighborhood in Croatia and have known each other for years. They still practice together on tour.
"We've practiced many, many times," Cilic said, later adding: "I think we both knew each other's games, maybe even more than enough on some occasions. It's tough to find the right solution."
Cilic said he had chances early in the first set to stay even with Dodig, but missed opportunities and a strong serve from Dodig early had him on the defensive until early in the second.
"He served extremely well, and I didn't have many chances on the return to make something," Cilic said of the first set. "Of the chances I got, I think he played extremely well, or I missed a couple of opportunities."
At that point, Cilic said he wanted to maintain his approach and handle the early play of Dodig. The match turned early in the second set.
"All and all, I played solid first set, except a couple of unforced opportunities on my serve," Cilic said. "I felt I just needed to try to keep playing like that and eventually, hopefully, the chance will be there.
"I felt like I was playing better and that he was dropping his level on his service game. That was the key in the second and third set where he didn't serve as well."