Miami topped 20 points for the first time since October but couldn't keep up with Beckham and the Giants.
With Miami's defeat, New England clinched the AFC East title.
"They got it to a star player,Wholesale Jerseys, and he made plays for them," Dolphins defensive tackle Ndamukong Suh said.
Beckham's final catch, a diving grab on third down at midfield, sealed the victory with less than two minutes left.

Tannehill threw for 236 yards and burned a blitz by throwing deep to Kenny Stills for a 47-yard score that put Miami ahead 24-17. But Tannehill overthrew an open Landry behind the Giants' secondary with his team trailing and less than five minutes to go.
The New York Giants rallied from a fourth-quarter deficit thanks to two late touchdown catches by Odell Beckham Jr., and Miami was eliminated from the playoff race with a 31-24 loss.
"You find out who loves the game, because that's what you're playing for," Campbell said.
"Obviously this is not where we want to be," quarterback Ryan Tannehill said, "but here we are."
"At the end of the day, we shot ourselves in the foot," receiver Jarvis Landry said.
"Obviously his performance is something we all expect," Landry said. "He's just continuing to do what he does."
He had seven catches for 166 yards and the game's final two scores. The 100-yard game was his sixth in a row, a Giants record.
Cornerback Jamar Taylor and seldom-used safety Shamiel Gary were the closest defenders, but Manning hit Beckham in stride at midfield and he sprinted to the end zone for his 12th TD this year.
The Dolphins were hurt by 12 penalties and seemed reluctant to run the ball, as has been the case most of the year. Lamar Miller rushed for 89 yards, including a darting 38-yard touchdown run after making Jason Pierre-Paul miss,Cheap Jerseys USA, but carried only 12 times.
The Giants (6-7) broke a three-game losing streak and are tied with the Redskins and Eagles for the lead in the woeful NFC East.
The Giants plotted the play at the team hotel before the game.
MIAMI GARDENS, Fla. (AP) — The Miami Dolphins introduced their 50 greatest players at halftime Monday night, and then the 2015 Dolphins showed why they're not so great.
In celebration of their 50th season, Miami honored its greatest players. The festivities seemed to inspire the Dolphins, who started fast, but were eventually run down by Beckham.
NOTES: Dolphins CB Bobby McCain went into the concussion protocol. He briefly lay motionless when he was injured covering the kickoff to start the second half, but walked off the field. ... The Dolphins fell to 41-41 on Monday Night Football.
"The corner was supposed to follow up and go over the top," Jones said, "which he didn't."
Miami safety Reshad Jones blamed miscommunication.
Eli Manning went 27 for 31 for 337 yards and four scores. His passer rating of 151.5 was his highest since 2009, and his completion percentage of 87 was a career best.
Beckham missed one series when he went to the locker room due to cramps in both calves, but didn't let that slow him down. And with the score 24-all, he took advantage of broken coverage to score the go-ahead touchdown.
"We got the coverage we wanted,Cheap NFL Jerseys China," Manning said. "We thought it would be a big play, but I didn't think he would be that open."
Beckham slipped 5 yards behind the secondary to make an 84-yard touchdown reception that put the Giants ahead to stay with 11 minutes to go,NFL Jerseys Outlet.
"Those are the ones you've got to make," Tannehill said.
The loss makes it less likely interim coach Dan Campbell will return next year. He fell to 4-5 since taking over, and his team must dig deep for motivation in the final three games.
The Dolphins (5-8) are assured of extending their postseason drought to a franchise-record seven consecutive years,Cheap Jerseys Free Shipping.
Beckham outplayed Landry, his close friend and former LSU teammate. Landry had 11 catches for 99 yards but was flagged for a costly personal foul penalty with his team trailing in the fourth quarter.