Showing no ill effects from a torn tendon just before the season's opening Grand Slam tournament, Agassi committed only six unforced errors none in the final set.
The 29th-seeded Dent rushed the net on 136 of the 201 points but constantly found himself lunging and diving for shots, and winning only 51% of the forays. The packed centre court roared in approval of the strong performance by both players.
"It's always a bit of a deceiving stat when you got a guy that's putting so much pressure," Agassi said. "It seems like any time I did miss, it was because he was putting pressure on me."
Dent is one of the few pure serve-and-volleyers left in tennis.
"If you're not on your game, he's one of the worst guys to play," said Agassi. "The guy's really talented, can make a lot of shots that you just don't expect him to be able to pull off, and he does them at the most important times."
Agassi was down 4-1 in the first set before coming back, although Dent still nearly forced a tiebreaker. Then Agassi went up a break early in the second set, only to see Dent tie it 3-3.
Dent fended off five break points while serving at 5-5, then squandered triple break point in the next game. At deuce, Agassi made a rare venture to the net and tapped a backhand drop volley that briefly rolled along the tape before falling for a winner.
Agassi never trailed in the tiebreaker, finishing it with a crisp backhand crosscourt pass. That seemed to take the steam out of Dent, who won only 11 points in the last set.
Agassi is one victory from a quarter-final with Federer. The top-ranked Swiss lost his first service game, then got back on track and advanced when Finland's Jarkko Nieminen quit with an abdominal muscle tear on a day when injuries played a role in several matches.
Agassi next faces another power player, No. 11 Joachim Johannson of Sweden, who survived a four-hour struggle against No. 24 Feliciano Lopez of Spain that went to 13-11 in the fifth set.
Playing better after injuring his right ankle, fourth-seeded Marat Safin was testy at times but kept his notorious temper in check in a 6-4, 3-6, 6-3, 6-4 victory against Croatia's Mario Ancic.
"When I twisted my ankle, I just kind of stopped a little bit thinking about the way I was playing," said Safin.
French Open champion Gaston Gaudio, seeded 10th, was able to continue after treatment on both thighs, but Dominik Hrbaty upset him in five sets despite 88 unforced errors.
In women's singles, second-ranked Amelie Mauresmo and US Open winner Svetlana Kuznetsova won in straight sets and will face unseeded opponents in the fourth round.
Serena Williams, seeded seventh, ended the run of wild card Sania Mirza, the first Indian woman to reach the second round of a Grand Slam. Williams defeated Mirza 6-1, 4-4, ending with a second-serve ace and skipping off the court. She next meets No. 11 Nadia Petrova.
"Finally, after forever it seems, (I was) able to feel my serve," Williams said. "So I was really happy about that."
Wimbledon winner Maria Sharapova beat China's Li Na 6-0, 6-2. Li had only three winners in her 50-minute loss. Sharapova, one of three Russians among the top six seeded players, next meets 15th-seeded Silvia Farina Elia of Italy.
Farina Elia cruised past British qualifier Elena Baltacha 6-1, 6-0.
"I'm a much better player than I was seven months ago," said Sharapova.
Mauresmo defeated Serbian teenager Ana Ivanovic 6-2, 7-5 and awaits Evgenia Linetskaya, a 6-3, 6-4 winner over American Amy Frazier.
"It's a long way still three matches down, four to go and some good players still in the field," said Mauresmo said, referring to prospects for her first Grand Slam tournament title.
Kuznetsova made 48 unforced errors in her 6-3, 7-6 (7-5) win over Mariana Diaz-Oliva.