Suddenly, the smooth-running, efficient penalty-killing machine known as the Kings started sputtering and eventually broke down at Staples Center.

It was supposed to be the Flames that were beset by woeful penalty killing. But the teams traded places as Calgary beat the Kings, 3-2, on Monday, scoring all three goals on the power play.

The final salvo was a painful one for the Kings, coming with 29.7 seconds remaining in regulation. Calgary defenseman TJ Brodie scored his first of the season, converting on a low shot from the top of the slot to beat Kings goalie Jonathan Quick.

GAME SUMMARY: Flames 3, Kings 2

Kings center Anze Kopitar had gone off for hooking Flames defenseman Dennis Wideman at 17:48 of the third period. In all, Calgary went three for six on the power play.

"I think maybe everyone is trying too hard," Kings center Mike Richards said of the penalty-killing effort. "When you over-work, I think when you are out-manned, you run out of position and they get some open looks. Tonight, a guy went over to the other side and passed right through me. Sometimes, being simple is the best method."

Said Kopitar: "We took some bad penalties, including myself. It was a stupid penalty on my part and shouldn't have happened."

Earlier, the great center scoring drought appeared to come to a merciful end for the Kings. But it didn't even last beyond the second intermission. The Kings are still without a goal by any of their centers after 10 games.

For a few fleeting minutes, Richards was credited with a short-handed goal late in the second period. But nearly everyone in the building — even those with poor eyesight in the cheap seats — thought the goal eventually would be changed to Jeff Carter.

And it was.

Carter made a superb rush down the right wing, victimizing Flames defenseman Mark Giordano and Carter's centering pass went off Wideman's stick, the deflection beating goalie Karri Ramo. It was Carter's fifth goal of the season.

Until then, it had been a battle of the team's respective power plays. The Kings had not given up a power-play goal since Jeff Skinner's goal late in regulation at Carolina on Oct. 11. Since then, they had killed 14 straight penalties.

Calgary ended that impressive run early in the second period and added another for good measure later in the period. The first came from a wide-open Mike Cammalleri in the right circle, at 2:50 of the second period, making it 1-1.

Cammalleri's one-timer was his first of the season, in his first game, as the former Kings forward has been out because of an injured hand.

Flames rookie center Sean Monahan capitalized on the power play at 13:39 of the second, giving Calgary a 2-1 lead after defenseman Drew Doughty had gone off for holding Sven Baertschi. Monahan was opportunistic around the net, scoring on the rebound, for his sixth goal of the season.

Monahan has scored in six of the Flames' eight games this season and the only team to hold him without a point happened to be the Ducks on Oct. 16

The Kings' only lead of the game came on Doughty's power-play goal at 18:36 of the first period.

lisa.dillman@latimes.com

Twitter: @reallisa