Having dramatically surrendered their lead with seconds remaining in regular time, the Rams gathered Bears quarterback Caleb Williams' third interception of the night, before Stafford stewarded his team into field goal range for a nerve-jangling victory.
It was a triumphant ending to a tough night for Stafford, in which the Most Valuable Player-tipped quarterback was bailed out by a Rams defense that stood firm in Chicago's 18 degrees F (-8 degrees C) snow blizzard, and Kyren Williams rushed for two touchdowns.
"It was crazy, it was back and forth," said Stafford. "We didn't play our best on offense, our defense dominated the game. It was unbelievable to watch."
"You'd love to come in here and throw for 450 (yards) and four touchdowns and do all that, but hey, find a way to win the game, survive and advance. And we did it."
A Chicago team so prolific at comebacks this season they have been dubbed the "Cardiac Bears" had forced overtime through quarterback Caleb Williams' outrageous backwards-sprinting blind pass with less than 30 seconds remaining.
On fourth down, Williams ran 20 yards away from goal before pivoting to hurl it at tight end Cole Kmet, who was standing inexplicably alone in the end zone, sparking raucous scenes at Soldier Field.
The Bears then prevented the Rams scoring on the opening drive of overtime, meaning Williams only needed to get into field goal range.
But the Chicago quarterback's wayward pass was scooped up by a diving Kam Curl.
Having been sacked four times, and managing only a sub-50 percent passing completion all night, Stafford expertly steered his own team into range.
Harrison Mevis converted a 54-yard field goal to set up an NFC Championship Game against the Seattle Seahawks.
Smothering defense
The Rams had reportedly arrived from sunny California with cayenne pepper in their socks to keep their feet warm for one of the coldest playoff games in Chicago football history.
After the Bears' bold attempted touchdown pass in an early 4th down was intercepted, the Rams scored on the ensuing drive, with running back Williams rushing for a touchdown.
Chicago's aggressive tactics were justified in the second quarter, as Williams successfully passed to DJ Moore for a touchdown on 4th down.
The game then settled into a war of attrition, with both defenses standing firm and explosive plays proving near-impossible in a frozen, scoreless third quarter.
Stafford repeatedly struggled to pick out his dangerous wide receivers - Puka Nacua and Davante Adams - as they were smothered by Chicago's secondary.
Williams' second rushing touchdown was canceled out by Kmet's last-gasp touchdown, setting the stage for overtime.