The 22-year-old Swiss, who lost in the second round last week in Rome, survived a tricky first-round match with flashes of brilliance and plenty of determination.
"I knew he was very good player and the conditions were tough with a heavy court and a lot of bad bounces but I hit the ball pretty well," Federer said.
"Those matches help a lot," he added. "This is good for my confidence for the rest of the tournament."
The Wimbledon and Australian Open champion, warming up for the French Open starting on May 24, needed just 28 minutes to take the first set, which he wrapped up with a forehand winner.
Gaudio, a semi-finalist here last year, woke up to open a 4-1 lead in the second set. Federer fought back to level at 5-5 but lost the set when he was broken in its final game.
The decisive set was tight until the Swiss all-court player, who made his breakthrough by winning in Hamburg two years ago, broke Gaudio in the ninth game to serve for the match.
Federer won his service game to love, sealing victory in just under two hours with an ace on his first match point.
Carlos Moya, the seventh seed and the winner of the Rome Masters on Sunday, underlined his great claycourt form with a 7-5, 6-4 victory over fellow Spaniard David Sanchez.
EARLY EXIT
Local hope Rainer Schuettler, the fourth seed, made an early exit, losing in three sets to fellow German Lars Burgsmueller.
Schuettler's friend and training partner, Burgsmueller won 6-4, 6-7, 6-4 after an exciting battle that thrilled a 9,000 crowd for more than three hours.
Former Wimbledon finalist David Nalbandian, the sixth seed, also went out. Spaniard David Ferrer crushed him 6-2, 6-3 and has now beaten the Argentine three times in as many meetings.
Lleyton Hewitt earlier advanced to the second round with a convincing 6-0, 7-6 victory over experienced Swede Jonas Bjorkman.
The Australian former world number one, seeded 17th in the claycourt tournament, was in a class of his own in the first set, which he took in just 22 minutes with an acrobatic backhand volley on set point.
The second set was tight and both players held serve until a tiebreak, which Hewitt took 7-5.
A service winner earned the former Wimbledon and U.S. Open champion a match point on which Bjorkman hit a backhand wide to bow out after 88 minutes.
"I played really well in the first set and dictated play from the start," said Hewitt, who next meets German Tommy Haas.
"I didn't feel I played badly at all in the second set. Jonas just picked up his game."
