Schenk birdied two of the last three holes in windy conditions to shoot a one-under par 70 and stand on eight-under overall after 54 holes at Innisbrook's Copperhead course in Palm Harbor, Florida.
Chasing his first PGA Tour title in his 165th start, Schenk fired his approach at the last inside five feet, then made the putt to seize the lead.
"Didn't have a ton go our way until the very end. Tough day but overall very pleased," Schenk said. "It was difficult. Wind was up, switching, it got cold at the end. We just stuck to our gameplan, kept our head down."
Schenk, playing for a 10th consecutive week ahead of becoming a father next month, has a career-best of third at the 2021 Shriners Children's Open but will be with three-time major winner Spieth in Sunday's final pairing.
"I just focus on myself," Schenk said. "I knew that last putt was to make it into the final group. Happy I made it. I'm excited for tomorrow."
Spieth, the 2015 Valspar champion, and Fleetwood, seeking his first PGA Tour win after 111 starts, each fired 69s to share second on seven-under with Webb Simpson, the 2012 US Open champion, with fellow Americans Cody Gribble and Taylor Moore on six-under.
"Didn't quite have my best stuff on the approach game and lag putting, but overall, I'm in a good spot going into tomorrow," Spieth said.
"Lot of good players up there. Tommy has been in good form... I still think after it rains tonight it's going to take some more scoring than the last couple days."
Fleetwood has won six European Tour titles, his most recent victory at last November's Nedbank Challenge in South Africa, but has four runner-up efforts without a victory on the US circuit.
"We all dream and every time we tee it up, every week we want to win," Fleetwood said. "I haven't done it yet. We can start at any time and start racking them up.
"Happy to be in contention again... overall happy with where I feel like my game is heading. Just try to do all the right things tomorrow and see if it's my time or not."
Fleetwood had a tap-in birdie at the par-5 first and sank another birdie putt at the par-5 14th while making 16 pars.
"Started off with a nice birdie and it was like keep putting away," said Fleetwood. "Birdies were very hard to come by."
SPIETH ROLLERCOASTER
Spieth endured a roller-coaster day with five birdies, five bogeys plus an opening eagle.
A closing bogey cost him a share of the lead heading into the final day.
"Eventful," Spieth said. "For the amount of mistakes I made, it's nice to shoot a couple under."
World number 14 Spieth is seeking his 14th PGA Tour title and his first since last April's Heritage event.
Spieth dropped his 6-iron second shot eight feet from the opening hole and sank the eagle putt to immediately tie the lead.
"Great way to start, especially on a day where you know you're going to get a lot of wind," Spieth said.
"Par-5s were great. That's what you need to do in these kind of conditions on this golf course."
He made a birdie at the par-5 fifth, then had four bogey-birdie runs between the seventh and 16th holes.
"I tried to be a little more aggressive with the putter," Spieth said.
