The Associated Press

KANSAS CITY, Mo. — The Kansas City Chiefs gave Dick Vermeil a near-perfect goodbye gift.

They couldn’t give him what they really wanted, though — one more game as their head coach.

The Chiefs were unstoppable on offense and stifling on defense in Sunday’s 37-3 victory over the playoff-bound Cincinnati Bengals. But Pittsburgh’s 35-21 victory over Detroit kept the Chiefs (10-6) from reaching the AFC playoffs, meaning Vermeil’s decision to retire at the end of the season takes effect immediately.

‘‘I think they’ve demonstrated that they are a playoff-caliber football team,’’ Vermeil said. ‘‘We just didn’t get some things done that we had to get done at the time we needed to get them done.’’

Cincinnati, its playoff berth already in hand, didn’t need to get a lot done on Sunday — and it didn’t.

Only Shayne Graham’s first-quarter field goal kept the Bengals (11-5) from being shut out for the first time since 2001, and the AFC North champions take a two-game losing streak into their first playoff appearance in 15 years.

But at least they get to go, as the No. 3 seed hosting Pittsburgh in the first round Sunday. Cincinnati split the regular-season series with its AFC North rival, each team winning on the road.

‘‘This game is over with, and we’ve got to move on,’’ Bengals defensive end Justin Smith said. ‘‘It really doesn’t mean anything. There’s no excuses, but we knew we were going to play next week. It’s win or go home now.’’

Kansas City becomes only the fourth team to go 10-6 and miss the postseason since the NFL expanded the field to 12 teams after the 1990 season.

‘‘It’s difficult. I’ll be honest with you,’’ guard Brian Waters said. ‘‘I feel like we left a lot of things out on the field as an offense, but all you can do is finish the best you can. The team didn’t quit. We gave it a shot and put a lot of pressure on the other teams.’’

The Chiefs’ offensive successes down the stretch — especially the emergence of Larry Johnson as one of the league’s elite running backs — makes missing the playoffs even harder to take.

Johnson ran for 201 yards and a career-high three scores on 21 carries, his ninth straight 100-yard game, and broke Kansas City’s single-season rushing record.

Johnson, who scored two touchdowns 40 seconds apart late in the second quarter, finished with 1,750 yards rushing to break Priest Holmes’ club record of 1,615 yards — despite not becoming the regular starter until Nov. 6, after lingering effects of head and neck trauma ended Holmes’ season.

‘‘I feel like I could play 10 to 15 more games, because I’m so young,’’ said Johnson, a third-year professional who will play in his first Pro Bowl next month. ‘‘People think that because I have a lot of carries, I’m taking a lot of shots, but I’m not.’’

His streak of nine straight 100-plus rushing games is the third longest in NFL history, tied with Walter Payton, Fred Taylor and Deuce McAllister.

He just missed a fourth touchdown midway through the fourth quarter when he fumbled at Cincinnati’s 2-yard line.

Rudi Johnson, held to 18 yards on 10 carries, still broke his single-season rushing record for Cincinnati. He finished with 1,457 yards, three more than his previous mark.

But with his struggles, and a relatively quiet day for irrepressible Bengals wide receiver Chad Johnson, there was no question which Johnson was the game’s biggest star.

With the Chiefs up 6-3 late in the second quarter, Larry Johnson broke through the line and ran 49 yards for the game’s first TD. Three plays after that, tight end Matt Schobel tipped Jon Kitna’s pass and Sammy Knight intercepted at the Bengals 41.

After Trent Green’s 27-yard pass to Tony Gonzalez, Johnson started left, spun out of David Pollack’s tackle and cut back across the field — picking up a key block from Green along the way — for a 14-yard TD.

It was the first of two interceptions thrown by Kitna, who replaced Carson Palmer to start the second quarter.

After Johnson’s 20-yard run on the opening drive of the second half gave the Chiefs a 27-3 lead, Patrick Surtain’s interception set up Lawrence Tynes’ 23-yard field goal.

Palmer, who sustained a groin strain last week against Buffalo, said the injury wasn’t an issue.

‘‘My groin felt great,’’ he said. ‘‘I could have ended up playing the whole game. ‘‘But I think Coach (Marvin Lewis) wanted to let some of the starters rest. Looking back, I’m glad I let my groin rest.’’

Tynes also kicked first-half field goals of 39 and 24 yards for the Chiefs.

Green finished 23-for-29 for 335 yards, closing out Kansas City’s scoring with an 8-yard pass to Dee Brown.

Eddie Kennison had 151 yards on seven catches for the Chiefs.

Chad Johnson was held to 55 yards on four catches.

Notes: Rudi Johnson and Chad Johnson are only the third set of teammates to set franchise rushing and receiving records in the same season, joining Dallas’ Emmitt Smith and Michael Irvin in 1995, and Houston’s Domanick Davis and Andre Johnson in 2004. Chad Johnson extended his record to 1,432 yards receiving. … Gonzalez set a team record with his 84th straight game with at least one reception. He had shared the record with Stephone Paige (1985-91). … Graham’s 49-yard field goal in the first quarter was his 14th straight made attempt, a Bengals record.