Furyk, the 2003 US Open winner, carded an eagle, seven birdies and two bogeys to sit atop the first day's leaderboard at seven under par.
The American's nearest challenger was Slattery, who signed for a 65 following a remarkable run around the turn.
Like Furyk, Slattery started on the back nine and, after beginning with six consecutive pars, he then recorded six birdies and a bogey in seven successive holes to fly up the leaderboard before finishing his round with five more pars at the Firestone CC (South Course) in Ohio.
Slattery's fellow Englishmen Luke Donald and Simon Dyson were part of six-man group on four under par, along with Americans Bubba Watson and Ben Crane, Spaniard Rafael Cabrera Bello and John Senden of Australia.
World number one Donald, looking to put his disappointing showing at The Open behind him, made a blistering start with birdies on his first three holes before dropping shots at the 13th - his fourth - and 18th to slip back to one under.
He got back on track with four more birdies on the front nine before disappointingly dropping a shot on his penultimate hole.
Further down the field, Lee Westwood was on two under after a 68 while Tiger Woods and Rory McIlroy, number two and three in the world respectively, were on level par after shooting 70s.
Woods, a seven-time champion at Firestone, had looked set for a much better start to the tournament after reaching three under after 11 but bogeys at the 13th, 16th and 18th saw him slip back.
Open champion Ernie Els, meanwhile, was three over after posting a 73.
Watch: Furyk takes early Firestone lead... More Videos
Furyk has had a disappointing time of it since the US Open in June, when he finished tied for third having been joint leader heading into the final round.
In his four tournament outings since then, the 42-year-old has finished tied for 34th twice and missed two cuts, the second of which was at the Canadian Open last weekend following two rounds of 70.
After that he took the chance to fly home to Florida for a few days - something which appears to have worked well for him.
"I think more than anything I needed a little time to clear my head, think about what was going wrong," said Furyk, whose 63 today was one shot off his career low and two off the course record.
"It wasn't anything that was going wrong, why I wasn't playing better. I just felt like I needed to come in here and quit concentrating on trying to be so mechanically sound and just go play some golf and try to score and get the ball in the hole a little bit. It worked today. I did a lot better job of scoring.
"It's been a while since I made seven birdies and an eagle in a round, so it was a lot of fun," he added on www.pgatour.com.