Kyle Capobianco scored in overtime for the Coyotes (4-13-2), who have won consecutive games for the first time this season and are 3-0-1 during a four-game point streak.
Vejmelka was 0-9-1 in his first 12 games (10 starts) this season.
"We felt for him a little bit because he was so good early in the year, was by far our best player, and we could not score the big goal," Arizona coach Andre Tourigny said. "The game we won tonight, we lost them [previously], so he was right there. He was doing miracles, but we could not get the goal to make him get a reward."
It was the second game of a back-to-back for Arizona. They won 2-1 in overtime against the Detroit Red Wings on Saturday in nearly identical fashion, including scoring the tying goal in the third period.
"We know we're not out of games," Capobianco said. "We're just sticking with our systems right now, wearing teams down and coming back in the third. And I think that just shows how committed we are to our systems and sticking with it."
Brendan Lemieux scored for the second straight game for the Kings (8-7-3), who are 0-2-2 following a seven-game winning streak. Jonathan Quick allowed two goals on 27 shots.
"Tonight is a frustrating night because of the amount of chances that we created," Los Angeles coach Todd McLellan said. "We'd probably be talking a different story if we had some finish, but we didn't so tonight's a tough one to evaluate. The chances were there, so it's disappointing."
Capobianco's shot from the right face-off circle went off the crossbar and in at 3:46 of overtime. Play continued afterwards briefly as the Kings believed the puck had not entered the goal.
Capobianco started celebrating by embracing Clayton Keller, who had the primary assist. However, Capobianco got worried when Kings captain Anze Kopitar started down the ice the other way on a breakaway.
"My heart kind of skipped a beat there, and I just tried to skate as hard as I could back. Luckily, it was in, and we could celebrate after that," Capobianco said.
It was Capobianco's third goal in 19 NHL games and his second this season.
"He has the poise, that swagger to make big plays," Tourigny said. "So I think he needs to keep working on his all-around game, but the feel is there. The talent is there. It's a matter now to keep working with the young guy and developing him."
Travis Boyd was at the foot of the crease to put in Shayne Gostisbehere's pass and tie it 1-1 at 4:44 of the third period.
