PALM BEACH, Fla. — The N.F.L.’s 32 clubs passed a rule change on Tuesday to ensure that both teams would possess the ball at least once in overtime of postseason games. The measure comes months after Kansas City won a divisional round playoff game against the Buffalo Bills, who were not given a chance to score in overtime.

The change in the league’s overtime rules was their first since 2010, when clubs voted to allow teams that scored a touchdown on the opening possession of overtime in a playoff game to win. (Before that, the team that scored first in any way in overtime won.) The rule, which by its nature gave an advantage to the team that won the overtime coin toss, was extended to the regular season in 2012.

Since 2010, there have been 12 postseason overtime games, and the team that won the coin toss preceding overtime went on to win 10 of those 12 games. Seven of those 10 wins came on a first-drive touchdown.

The game between Kansas City and Buffalo in January might have been the most dramatic of all such games. The two teams’ high-powered offenses scored a total of four touchdowns in the final two minutes of regulation, and Kansas City won, 42-36, by scoring a touchdown on the first possession of overtime.

In the aftermath, commentators, fans and football executives lamented that viewers did not get to see the Bills and quarterback Josh Allen try to score in overtime, too, a furor that catalyzed the league to adopt the new rule.

“There has to be the latest example for change, and that was the last straw that now, hey, we need to move forward and do this,” said Bills Coach Sean McDermott, who called the rule change “bittersweet.” “It’s the right thing for the game.”

Both teams will now be guaranteed at least one possession, regardless of the clock. If both teams score a touchdown on their opening drives, then the team that scores next wins.

The proposal for the rule change, which was made by staff members of the Indianapolis Colts and Philadelphia Eagles, would have affected both the 2021 regular season and postseason. McDermott said there was strong support for making this change only for the postseason, explaining, “That is where we were going to start.”

The rule change, however, will extend games. Some coaches, including John Harbaugh of the Baltimore Ravens, flagged it as a player safety concern, a reason for limiting it to the postseason, when teams’ seasons are on the line.

Kansas City proposed overtime changes after the 2018 season that would have given both teams a possession, after it lost the A.F.C. championship game to the New England Patriots, who scored on a first-drive touchdown.

McDermott said the Bills’ playoff game against Kansas City was cited several times in the meeting as teams discussed the rule change.

“It’s potentially the greatest 20 to 30 minutes of football that I’ve ever seen,” said Rich McKay, the Atlanta Falcons’ president and the chairman of the N.F.L.’s competition committee. “And to think it ended that way definitely brought up the idea of, hey, does that work for everybody?”