Fighting through a hand injury that appeared to dramatically impair his passing, quarterback Stafford threw a game-saving touchdown in the final minute to spare the Rams' blushes and advance to the divisional round.
"Total team effort. Obviously had some mistakes along the way. I turned it over," said Stafford.
"So a lot to clean up. But we found a way to win it late," he added.
The Rams had entered what proved a nervy, unpredictable and error-strewn clash as overwhelming favorites against a Carolina team that reached the playoffs despite finishing the regular season with an 8-9 losing record.
And the Los Angeles franchise started the NFC's first post-season clash with surgical precision.
Star wide receiver Puka Nacua grabbed all three passes of Los Angeles' opening drive, including a 14-yard touchdown reception.
He then received a lateral pass from Stafford and rushed five yards for his second touchdown and a 14-0 lead.
But the Panthers immediately fought back.
A seven-play, 65-yard drive from kickoff ended with a rushing touchdown for Chuba Hubbard.
The Rams grew nervy, spurning chances to build their lead. Suffering a blow to his hand in the second quarter, Stafford missed seven passes in a row. Nacua dropped a touchdown opportunity.
Quarterback Bryce Young - chosen by Carolina first overall in the 2023 draft - showed his athleticism in a 16-yard scramble for a touchdown just before half time.
With pressure mounting in the second half, Stafford saw a pass intercepted, and the Panthers took their first lead of the night with another Hubbard rushing touchdown.
The Rams regained their composure, with Stafford finding Kyren Williams for a no-look TD pass.
But there was a dramatic twist ahead. With barely four minutes left, the Panthers blocked a fourth-down punt to regain possession at the Rams' 30-yard line.
Young found wide receiver Jalen Coker for the go-ahead touchdown.
Finally, Stafford again pulled it back, locating Colby Parkinson for a 19-yard game-winning touchdown.
"I believe in my teammates. I know they believe in me," said a relieved Stafford.
Stafford told reporters that the bent finger he suffered was "not pleasant."
"We'll see what it is. I was obviously able to finish the game and throw a decent (pass)," said the 37-year-old Most Valuable Player frontrunner.