On a wild day with 12 games, Green Bay set up a showdown with Detroit next weekend to decide a division crown while Cleveland avoided a winless season and San Francisco snapped a 13-game skid.
Tennessee quarterback Marcus Mariota and Seattle receiver Tyler Lockett suffered broken legs, dimming title hopes for their squads, but the biggest loss was Carr.
Oakland (12-3) stayed on the heels of New England (13-2) for a home playoff edge by downing Indianapolis 33-25, but the Raiders were undone when quarterback Carr was injured when sacked early in the fourth quarter by Colts linebacker Trent Cole.
"A good win but a tough setback to have our QB go down like that," said Raiders coach Jack Del Rio, who added that Carr will have surgery Sunday on his fractured right fibula.
Matt McGloin, 1-5 as a starter (all in 2013) with only 13 career NFL games, will replace Carr in hopes of keeping the Raiders' title bid on track.
Dallas, Seattle, New England and Oakland had already secured playoff berths and New Orleans beat Tampa Bay 31-24 to ensure the Giants a wildcard spot and Atlanta the NFC South division crown.
Houston clinched its place in the knockout fight to the Super Bowl with a 12-10 victory over Cincinnati that wasn't secured until Bengals kicker Randy Bullock was wide right on a 43-yard field goal attempt on the final play.
Aaron Rodgers completed 28-of-38 passes for 347 yards and four touchdowns to power the Green Bay Packers over visiting Minnesota 38-25 for their fifth win in a row.
The victory means the Packers (9-6) will play for the NFC North title at Detroit on January 1 no matter how the Lions (9-5) fare at Dallas on Monday.
"We had some adversity and we stuck together," Rodgers said. "I'm really proud of the character of the football team. It has been one of those trying seasons where it hasn't quite come together at the right time.
"But over the last five weeks, we've shown a lot of mental toughness, a lot of grit, and guys sticking together when there could have been some fracturing."
The Vikings, who opened the season 5-0, fell to 7-8 and out of playoff contention after the team plane skidded off a runway after landing Friday at a nearby airport, stranding players and coaches on the plane for more than four hours.
"I don't think the plane ride had anything to do with it or anything else. There are no excuses," Vikings coach Mike Zimmer said.
Dolphins on playoff brink
Atlanta's Matt Ryan threw for 277 yards and two touchdowns while Matt Bryant kicked four field goals as the Falcons (10-5) beat host Carolina 33-16.
Miami's Jay Ajayi ran for 206 yards and a touchdown in the Dolphins' 34-31 victory at Buffalo. Andrew Franks kicked the winning 27-yard field goal with 47 seconds remaining in overtime.
The Dolphins (10-5) would claim their first playoff berth since 2008 if Kansas City beats Denver on Sunday.
New England routed the New York Jets 41-3 and can clinch a home-field playoff edge by beating Miami next week. The Patriots -- on a six-game winning streak -- gave coach Bill Belichick his 200th career victory as Tom Brady threw three touchdown passes, LeGarrette Blount ran for two touchdowns and Stephen Gostkowski kicked his 300th career NFL field goal.
Tennessee's playoff hopes ended due to the 38-17 loss at Jacksonville in which Mariota was injured.
Washington sustained wildcard playoff hopes with a 41-21 victory at Chicago. Redskins quarterback Kirk Cousins ran for two touchdowns and threw for another as Washington (8-6) enters next week's finale against the Giants with a chance to advance.
Browns won't go winless
The Browns avoided a winless campaign with a 20-17 home victory over San Diego, ensuring they will not match the 2008 Detroit Lions as the only 0-16 teams in NFL history.
San Diego's Josh Lambo missed a 45-yard field goal attempt on the final play to preserve Cleveland's long-awaited victory.
San Francisco snapped a 13-game losing streak by edging Los Angeles 22-21 as Colin Kaepernick flipped a touchdown pass to Rod Streater with 31 seconds remaining then ran in a game-winning two-point conversion. The 49ers had not won since beating Los Angeles in the season opener.