Budaj posted his NHL-leading seventh shutout, Carter scored midway through overtime and the Kings won their fifth in a row, beating the Philadelphia Flyers 1-0 on Saturday.
Budaj, who wasn't even on the Kings' roster on opening night, extended his shutout streak to 144 minutes, 5 seconds. He made 17 saves.
"To be honest with you, if he wasn't playing how he is, we wouldn't be in the position we are," Carter, who began his career with the Flyers, said of Budaj. "All the credit to him. He's been holding it together for us."
Anze Kopitar set up the winning goal at 2:35 with a backhand pass across the crease to Carter, who whizzed a wrist shot off Michal Neuvirth and into the net. It was the 27th tally of the season for Carter, who entered trailing Pittsburgh's Sidney Crosby by two goals for the NHL lead.
"I kind of got lucky," Carter said. "It hit off his blocker and went in. At that point, I'll take it."
Neuvirth made 27 saves for Philadelphia, including a sensational glove stop in the second period.
Budaj is filling in for starting goalie Jonathan Quick, who hasn't played since sustaining a groin injury on opening night. Quick, signed through 2023 on a 10-year, $58 million deal, has practiced with the team the last two days. But Budaj's play has allowed Quick to take his time in his recovery.
Budaj improved to 25-14-3.
"I feel healthy and strong," Budaj said. "Most of all, I think the team is playing really well in front of me. I'm enjoying it; this is great. It's amazing and I'm very excited and very thankful for this. So far it's been a great year for me. There's still a long way to go. I've got to take it one game at a time, keep working hard and help the team win."
The Kings goalie didn't have to make any spectacular saves in the old-fashioned defensive battle that featured lots of blocked shots, tight checking and strong goaltending from both clubs.
Tyler Toffoli nearly gave the Kings the lead four minutes into the second period, but his wrist shot from a sharp angle caught iron. Philadelphia's Sean Couturier got free for a breakaway 30 seconds later, but he also hit the post after beating Budaj.
Neuvirth kept the game scoreless with the save of the game, stopping Dustin Brown with 12:05 left in the second period after a giveaway by Philadelphia's Jakub Voracek. Brown had a wide-open net to fire into, but Neuvirth sprawled to his left with a fully extended glove and made an unlikely save.
"It kind of happened quick, and it caught me off-guard a little bit," Neuvirth said. "I just stayed with it, had my eye on it the whole time and it was a big save for us."
The Philadelphia goalie made another strong stop during a tight third period, denying Tanner Pearson's backhand try with a glove save with 4:24 left in the period.
Radko Gudas provided the hit of the game midway through the first period when he checked Kevin Gravel over the boards into the Flyers bench, where backup goalie Steve Mason had to brace himself for the incoming Gravel.
Philadelphia coach Dave Hakstol benched 23-year-old defenseman Shayne Gostisbehere and 19-year-old forward Travis Konecny for the second straight game. The young, promising duo can provide offensive spark, but Hakstol would like to see each improve his defense.
Philadelphia allowed a season-best 16 shots in Thursday's 3-1 home win over Montreal without Gostisbehere and Konecny in the lineup.
As in Thursday's victory, Hakstol liked his players' effort in the overtime defeat.
"I thought from the drop of the puck we worked hard, we battled pretty hard together," he said. "We played a pretty good hockey game. Guys left it all out there. Top to bottom, it was a good effort."
The Flyers coach was noncommittal as to when or whether his two young players would return to the lineup.
"I'm trying to dress the best roster possible to win hockey games; point blank," he said. "I haven't considered that. We'll consider that after we look at our performance tonight."
Flyers D Michael Del Zotto (lower body) missed his sixth straight game. ... Los Angeles F Jordan Nolan was not in the lineup after sustaining a lower body injury in the second period of Wednesday's 5-0 win over Colorado. ... Gostisbehere has been a healthy scratch in four games this season while Konecny has sat three times. ... Toffoli (lower body injury) returned to the lineup for the first time since Dec. 20. ... Drew Doughty got a secondary assist on the goal, extending his career-best streak to seven straight games with an assist. . The Kings improved to 11-4 in overtime.
Kings: Play the second contest of a four-game road trip at Washington on Sunday afternoon.
Flyers: Host St. Louis on Monday night.