Sanchez threw a pair of TD passes and three interceptions in his first start replacing injured Sam Bradford, but Chip Kelly's offense couldn't keep up with Winston and the Buccaneers.
After Lavonte David returned a fourth-quarter pick 20 yards for a score, Sanchez threw his hands up and shouted at Sproles because he stopped his pattern short.
"We got that cleared up quick," Sanchez said. "We're all good. He's a total pro and he's a great player."
The Eagles (4-6) have lost two in a row, but are only one game behind the New York Giants (5-5) in the mediocre NFC East. They'll have a quick turnaround with a game at Detroit (3-7) on Thanksgiving.
"You can't dwell or you'll keep getting kicked in the teeth," right tackle Lane Johnson said.
Winston completed 19 of 29 passes for 246 yards and had four of his TD passes in the first half to four different receivers. He tied Matthew Stafford's rookie mark in the third quarter on an 8-yard toss to Cameron Brate, the fifth receiver to catch a TD.
Martin had 177 yards in the first half, including runs of 84 and 58 yards. He finished 2 yards short of tying Barry Sanders' record for most without a TD. Sanders had 237 for the Lions against Tampa in 1994.
Tampa set a franchise record with 283 yards rushing and 521 total yards and were second-most in franchise history in the regular season.
The Buccaneers (5-5) have won two straight games for the first time in two years.
Winston and the Buccaneers were outstanding on third downs, converting 9 of 12 while building a 35-14 lead. Winston's last three TD passes came on third downs.
"A game like this will shake your confidence and that's our biggest enemy because it can destroy you," Eagles defensive coordinator Bill Davis said.
Winston threw a 17-yard screen pass to Charles Sims on third-and-6 to extend another drive, hit Evans for 25 yards and connected with Vincent Jackson over the middle for a 13-yard TD on the first play of the second quarter.
Martin later burst up the middle for 84 yards to the 1. On third down from the 4, Winston found Russell Shepard in the back of the end zone for a 21-7 lead.
The Eagles answered quickly as Sanchez hit Sproles for a 35-yard TD on a screen pass. But the Buccaneers came right back. Martin ran for 3 yards on fourth-and-1 and Winston tossed a 14-yard pass to Sims for a 28-14 lead.
Winston's pass to Brate made it 35-14.
That gave the Eagles more takeaways by their offense in the past two games than by their defense.
One play after running for 11 yards to the Buccaneers 39, Sanchez hit Josh Huff across the middle. Huff made a couple nifty moves and ran in to give the Eagles a 7-0 lead.
The Eagles punted on their next four drives.
"We didn't play well in any facet of the game and there's no excuse for that," Kelly said.
NOTES: Eagles TE Zach Ertz suffered a concussion. ... Eagles also didn't have RB Ryan Mathews. ... Winston was first QB to throw four TDs in first half vs. Eagles since Eli Manning in Andy Reid's last game as Philadelphia's coach in 2012. ... Sanchez has thrown an interception in six straight games.
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