The match was even until the 111th-ranked Horna broke Gasquet late in the first set. The top French player then unraveled, missing 15 forehands and committing 27 unforced errors overall.
"After losing the first set, I lost all my confidence," Gasquet said. "I played my worst match of the year."
The match lasted exactly an hour, and ended under light rain at the Foro Italico.
Canas has won three of his five meetings with Federer, all on hard courts. They have never met on clay.
Federer leads a field featuring the top nine players in the rankings, including three-time defending champion Rafael Nadal.
Nadal is off to another perfect start to the clay-court season, winning back-to-back titles at the Monte Carlo Masters and the Barcelona Open.
"Right now my confidence is very high," Nadal said. "But it's going to be very tough. It's my first time playing three weeks in a row on clay. Last year, there was a week between Barcelona and Rome."
The top eight seeded players have first-round byes in this tuneup for the French Open, which begins May 25.
Last year, Gasquet finished in the top 10 for the first time, won a career-high 49 matches and posted his best Grand Slam performance by reaching the Wimbledon semifinals. This year, he is the only player in the top 10 yet to reach a quarterfinal.
"I'm either way high or way low, and now I'm very low," Gasquet said, adding that he is not enjoying the sport right now.
"It's really difficult on the court. I need to win some matches and have fun."
"I knew he was capable, just today anyone could have beaten me," Gasquet said.