Stafford, who is in his first season with the Rams after spending his first 12 with the Detroit Lions, threw for 202 yards and two touchdowns and ran for another in the final game of NFL Super Wild Card weekend.
Superstar wide receiver Odell Beckham Jr., who signed midway through the season after leaving the Cleveland Browns, caught his first career playoff touchdown, while also throwing a 40-yard pass on a trick play to set up the Rams' fourth touchdown.
Rams receiver Cooper Kupp continued his excellent season with a touchdown catch as well as 61 yards receiving.
The Rams raced into a 28-0 lead in the first playoff game at SoFi Stadium, the site of Super Bowl 56 next month.
After the game, Rams head coach Sean McVay praised the play of Stafford upon his first career playoff victory.
"I thought he did a great job leading the way," McVay said. "He's still the same great player to me, and he always was before this game. I think it's good so you guys don't have to talk about that anymore. So pleased be able to do this with him. He's our leader out there, and we wouldn't be here without him."
The clash was the first NFL playoff game to be held on a Monday since 1988 and the first ever to be held on MLK Day.
While the Rams offense was purring throughout, their defense as a whole might have been the MVP on the night.
Cardinals quarterback Kyler Murray was under constant pressure throughout the game, throwing two interceptions.
One was returned by Rams linebacker David Long for a three-yard touchdown, the shortest pick-six in NFL postseason history.
Linebacker Von Miller racked up six tackles and a sack in his first postseason appearance since winning the MVP award at Super Bowl 50 with the Denver Broncos, while Rams safety Eric Weddle was playing in his first game since coming out of retirement, having last played in the 2019 season.
It was a meek way for the Cardinals' season to end, having been the final team to lose their unbeaten record earlier in the season and the first to reach 10 wins.
Having completed 19 of his 34 passes for 137 yards, with two interceptions and a 40.9 passer rating, Murray called the performance "disappointing."
"It's disappointing that we didn't make it a game and come out and play the football we know we're capable of playing, that's really the most disappointing part," Murray told reporters. "Losing is one thing, but when you don't even make it competitive, it's another thing.
"I put a lot on my shoulders, put a lot on myself and to dream of this moment and then be in the playoffs, the first game of the playoffs and then to play the way I did, to play the way we did, it's -- like I said -- disappointing."
The Rams will now travel to Florida to face the defending Super Bowl champions, the Tampa Bay Buccaneers, for a spot in the NFC Championship game.