The top-ranked Federer broke the record he shared with Bjorn Borg, the five-time Wimbledon champion who won 41 straight matches on grass from 1976-1981.
"It's nice to get any streak," Federer said. "I'm still going, so even better if I can maybe postpone it and make it even last longer. I'm surprised myself I've kept it that long. To come through today was my only wish, not to break the streak, but to have it come together is nice."
Federer led Gasquet 6-3, 1-2 on Monday when the match was suspended by rain. The pair returned to Centre Court under cloudy skies, and Federer needed only 37 minutes to finish off the 20-year-old Frenchman.
Federer closed out the match with an overhead smash, then smacked a ball into the stands and basked in a huge ovation. He left the court the same way he arrived -- donning a specially designed cream-colored sports jacket emblazoned with his name and three Wimbledon-style trophies on the breast pocket.
"It's very nicely done, only one of a kind," he said.
Federer's last loss on grass came in the first round of Wimbledon in 2002, when he fell to Mario Ancic in straight sets. His win streak includes three titles at Wimbledon and four at the warmup tournament in Halle, Germany.
Federer said he considers Borg's grass-court streak more impressive than his own because all the Swede's wins came at Wimbledon. Borg won five straight Wimbledons before losing in the 1981 final to John McEnroe.
"Wimbledon stays Wimbledon after all," Federer said. "The five Wimbledons and six finals is something almost beyond possibility for any player. For me, he (Borg) stays a hero."
Gasquet was coming off a victory at the grass-court warmup at Nottingham, England, but proved no match for the player who is an overwhelming favorite to become the third man in the Open era to win four straight Wimbledons. Borg and Pete Sampras (1997-2000) are the others.
Federer smacked 35 winners, to go with only 13 unforced errors. Gasquet had only seven winners and 11 errors. Federer converted five of seven break points, and faced only one break point -- which he saved in Tuesday's first game with a big forehand.
Hingis, back at Wimbledon for the first time since 2001, finished off a 6-2, 6-2 win over Ukraine's Olgo Savchuk. The match had been suspended by rain after the first set Monday.
Hingis won the last four games, mixing up volleys and drop shots with her all-court groundstrokes. She used a drop shot to set up a volley winner on match point.
Hingis, who won Wimbledon in 1997 at age 16, is continuing a comeback after three years off the tour because of a series of foot and ankle injuries and operations. Seeded No. 12, Hingis is one of only three women's champions in the field, along with three-time winner Venus Williams and 2004 champ Maria Sharapova.
Henin-Hardenne, who is seeking to add the Wimbledon title to complete a collection of all four Grand Slams, played the first match on Centre Court and needed only 56 minutes to dispatch Yuan Meng of China, 6-0, 6-1.
The third-seeded Belgian, coming off her third French Open victory and a win last week at Eastbourne on grass, was up 6-0, 4-0 before Yuan won a game. Henin-Hardenne had 28 winners, compared to just three for Yuan.
The first player to advance to the second round was fifth-seeded Svetlana Kuznetsova, the former U.S. Open champion, who beat Italy's Romina Oprandi 6-3, 6-2. She finished the match with an ace.
Men's fourth-seeded David Nalbandian, runner-up in 2002, crushed South Africa's Wesley Moodie 6-3, 6-4, 6-1 in just 1 hour, 37 minutes. He broke to finish the match, setting up a second-rounder against Arnaud Clement.
Eighth-seeded James Blake was a 6-3, 7-5, 5-7, 6-4 winner over Danish qualifier Kristian Pless.
Matches started on schedule Tuesday under cloudy skies, a day after rain severely disrupted the opening of the two-week championships. No matches were completed Monday, with 17 suspended and 47 postponed, meaning the All England Club will have to pay up to $1.8 million in ticket refunds.