Medvedev came through 4-6, 6-3, 6-4, 6-4 against the muscular 67th-ranked Hungarian, recording his best Slam performance this year after a third-round exit at the Australian Open and first-round loss at the French Open.
Fucsovics, a quarter-finalist in 2021, saw his faltering challenge unravel when broken early in the fourth set, before calling a medical time-out to treat a right foot injury.
Despite that setback, he still made Medvedev work for the victory, saving a clutch of match points before the third seed prevailed.
"It was a very tough match," said the 27-year-old former US Open champion. "Marton plays well on grass and has made the quarter-finals here, which I haven't done.
"I was more decisive after the first set, saved some break points and I'm happy to be through."
He added: "I want to do well here. It's my worst Grand Slam in terms of results so I have a big motivation to do well here."
"Even in Roland Garros I have managed to get into the quarters," the third seed told the Court One crowd, referring to clay which is considered to be his weakest surface.
"I have big motivation to change it so I hope I can show my best tennis and enjoy," added the 2021 U.S. Open champion who also made it into the last 16 in his last appearance here two years ago.
This season Medvedev has been on fire on the ATP Tour, winning five titles, but the Russian maverick has struggled to carry that form into the majors, falling early at both the Australian and French Opens.
For a while on Saturday, it seemed like he was heading for another speedy exit.
But Medvedev survived a wayward opening set to topple an opponent who had won their only previous meeting at a slam.
When the 67th-ranked Fucsovics gave Medvedev the run-around in the opening set, with the Russian misfiring his returns time and again, visions of his 2020 Roland Garros first-round win over the third seed must have flashed through his mind.
But Medvedev trampled on those dreams in the fourth game of the second set, when he broke Fucsovics to love after the Hungarian double-faulted to surrender his serve.
Playing under a closed Court One roof as the rain came down on leafy southwest London, both players entertained the crowd with some acrobatic shot-making.
Fucsovics would have made Boris Becker proud with a couple of his diving volley winners and slam-dunk smashes, while Medvedev hit a stupendous crosscourt winner on the run after chasing down a drop shot from well behind the baseline.
Once Medvedev had taken a two-sets-to-one lead by banging down an unreturnable serve, Fucsovics needed an injury time-out to get his right ankle manipulated and strapped up by the trainer.
He got back on his feet and even earned a break point in the eighth game of the fourth set. But once he missed his chance to make it 4-4, his game quickly unravelled and Medvedev reached the last 16 of a major for the first time this year by firing down an unreturnable serve.
"Very tough match. Marton plays well on grass and has played quarters at Wimbledon, which I haven’t yet done," said Medvedev.
"After the first set I was able to be a bit more decisive, save a couple of break points and I am happy to be through."
