"It's always nice to get the first match of the year out of the way," said the 20-year-old Cilic, a semifinalist at last year's Chennai Open. "What helped me was that I broke him early in both sets, because he then had to play catch-up after that."
Despite the relatively easy win, Cilic isn't ready to predict another spot in the semifinals.
"I'm not going to start thinking about making the final or winning the tournament. This is just the beginning and there are several good players in the draw," said Cilic, who won his first career ATP title last year in New Haven, Connecticut.
Schuettler, a finalist at the 1999 Chennai Open, broke Amritraj in the sixth and eighth games of the first set, but the 25-year-old Indian rebounded by winning the second set with a break in the third game.
"Prakash deserved to win the second set," said the 32-year-old Schuettler, who played more aggressively in the third. "I definitely need to pick up my game. It's not always smooth to get back competitive tennis after two months."
The Los Angeles-born Amritraj, whose tennis-playing father Vijay Amritraj moved from Chennai to California several decades ago, was back in the southern Indian city amid a debate over the government's recent decision to ban foreign-passport holders from representing the country in international events.
Amritraj, who holds a U.S. passport, has been a regular member of India's Davis Cup squad since 2003, but will now be affected by the Indian sports ministry's order.
Amritraj said he "won't comment on the issue at this stage."