The Packers, the youngest team in the NFL who snuck into the play-offs on the final day of the regular season with a 9-8 record, totally dominated the Cowboys before allowing two late touchdowns.
The Lions won a play-off game for the first time in 32 years with quarterback Jared Goff leading Detroit to a tight victory against his former team.
Packers quarterback Jordan Love, who took over in Green Bay this season after the departure of Aaron Rodgers, threw for 272 yards and three touchdowns in a stunning play-off debut.
Running back Aaron Jones had three rushing touchdowns and put up 118 yards on the ground and the Packers earned a clash with the top-seeded San Francisco 49ers.
Dallas gave up the most points in a play-off game in franchise history while quarterback Dak Prescott threw two costly interceptions in a nightmare first half which saw the home side fall 27-0 down in the second quarter.
The manner and scale of the defeat are likely to raise questions over the future of Cowboys head coach Mike McCarthy, a former Super Bowl winner with the Packers who was hired to end the wait for another Cowboys NFL championship, which has now stretched to 28 years.
Green Bay's confidence was evident from the outset as they elected to receive and scored a touchdown on their first drive - Jones rushing in from three yards.
Miserable Day
Prescott's miserable day began with an interception - cornerback Jaire Alexander picking him off and the Packers taking over 19 yards out and Jones scoring from three yards on the third play.
Green Bay's offense was meeting no resistance as Love delivered a 20-yard touchdown pass to Dontayvion Wicks who had escaped Dallas corner Stephon Gilmore.
It got worse for the Cowboys when Prescott's pass was grabbed by Packers safety Darnell Savage who returned it 64 yards to make it 27-0.
That was the largest deficit the Cowboys had ever faced at home in the post-season but they kept a glimmer of hope for the second half with Prescott finding Jake Ferguson with a one-yard pass in the final seconds of the second quarter.
After a Cowboys field goal though, Jones ran in his third touchdown, his nine-yard rush making it 34-10.
Tony Pollard ran in a touchdown for Dallas but again the Cowboys' defense couldn't stop Green Bay as Love found a wide-open Luke Musgrave with a 38-yard touchdown pass.
The unflappable Love found Romeo Doubs with a three-yard touchdown pass to give the visitors a 32-point lead with 10:23 left in the fourth quarter.
The Packers finally eased off, allowing two touchdowns for Dallas - both from Prescott passes to Ferguson but it was too late for any miracle comeback.
The Lions were hosting a play-off game for the first time since 1993 and they will be at their Ford Field again next week to face the winner of Monday's game between the Philadelphia Eagles and the Tampa Bay Buccaneers.
As well as pitting Goff against his former Rams, Sunday's game brought LA's quarterback Matthew Stafford back to face Detroit after the teams swapped the pair in 2021.
The Lions struck first in an explosive first half with David Montgomery rushing in from one yard for a touchdown and after a Rams field goal, the Lions opened up a 14-3 lead on Jahmyr Gibbs' 10-yard run.
Stafford brought the Rams back into contention with a perfectly delivered 50-yard touchdown pass wide-left to Puka Nacua.
Sam LaPorta collected a two-yard touchdown pass from Goff to restore the 11-point advantage for Detroit only for the Rams to respond swiftly with Stafford connecting with Tutu Atwell on a 38-yard touchdown.
The Lions extended their lead to 10 points with a 54-yard field goal from Michael Badgley but two field goals from Brett Maher brought the Rams within a point.
Detroit had the ball on their 17-yard line with 4:07 left in the game and Goff made sure of the win with two first downs, the second coming via a well-worked 11-yard pass to receiver Amon-Ra St.Brown which set off emotional celebrations among their long-suffering fans.
"It means a whole lot to this city, we knew what it meant when this season started, to get into the play-offs and then get this win," said Goff.
"It's just the beginning for us, we've got some road left."
