The Bucs' Super Bowl defense looked set to begin with a loss as they trailed 29-28 with seven seconds remaining in Thursday's opening-night thriller at Raymond Jones Stadium.
But inspired by seven-time Super Bowl winner Brady, the hosts edged a see-saw battle 31-29 thanks to Succop's game-winning kick in the dying seconds.
After surviving a scare against the Cowboys, Brady accepts there is plenty of work for his side to do.
"The margin of error is thin in the NFL," he said. "One or two plays, it's always that's the way the game goes. Fortunately, we found a way to win.
"I'm really happy Ryan and the field goal team made that really clutch kick there at the end. That was great to see. But as a team, we all know we've got to get back to work.
"We won, it's great, but we know that it was far from perfect and we have to get back to work and clean a lot of stuff up."
Brady completed 32 of 50 passes for 379 yards against the Cowboys and passed Drew Brees for most career passing touchdowns in season openers in NFL history with 43.
Cowboys counterpart Dak Prescott, making his comeback from a long lay-off, was 42-for-58 passing, with 403 yards, three TDs and one interception.
Prescott has now had 400-plus yards in four games since the start of last season – the most by any player in the NFL.
But from the Bucs' perspective, it was far from a vintage performance as they committed 11 penalties for 106 yards and converted just two of six third-down tries.
"We've got a lot a lot to learn," head coach Bruce Arians said. "I'm obviously not pleased with the start of the game, though I loved the finish.
"Our guys are winning. They're going to finish and we're going to win, but we can play better and not put ourselves in that situation."