After Tuesday’s morning skate, Flyers coach Alain Vigneault raised some eyebrows when he said the Buffalo Sabres had a “highly skilled team.”

He even managed to keep a straight face.

Buffalo has the fewest points in the NHL and has become the definition of futility. And the lineup it put on the ice Tuesday night was worse than usual because it didn’t include star center Jack Eichel, who missed the game with a suspected neck injury.

The Sabres, who took a seven-game losing streak into the night, are not a very good team.

» READ MORE: With Flyers almost healthy, Alain Vigneault looking for stability among lines and pairings

They blew a 4-2 lead as the Flyers stormed back and outlasted the Sabres in a shootout, 5-4, before 2,838 fans at the Wells Fargo Center.

Sean Couturier and Nolan Patrick scored in the shootout, and Brian Elliott denied both Buffalo shots.

“We faced adversity head-on, and found a way to win,” Vigneault said.

Shayne Gostisbehere, taking a slick drop pass from Couturier, scored on a power-play blast with 5 minutes, 46 seconds left to knot the score at 4-4.

Before that tally, the Flyers dominated the third period, but goalie Jonas Johansson had all the answers. And when he didn’t, he was saved by the goalpost (three times in the game), including Couturier’s long shot that was deflected by Joel Farabee with 8:06 left in regulation.

The Flyers killed a late tripping penalty on Jake Voracek, and the teams went into overtime.

Couturier (two assists) was stopped on a breakaway with 4:08 left in overtime. With 1:51 remaining, Elliott denied Buffalo’s Taylor Hall from point-blank range.

Elliott stopped 11-of-12 shots — the goal he allowed was scored on a shot that deflected off Gostisbehere — in relief of the struggling Carter Hart, who allowed three goals on eight shots and was removed after the first period.

“Moose came in and did a great job,” Hart said.

Hart wasn’t helped much by his teammates. Their defensive coverage down low was again a problem. It has been all season because the Flyers never replaced the retired Matt Niskanen with a reliable defender.

Afterward, Hart sounded as if his confidence was shattered.

“Just trying to stop the puck, and right now I’m not. It’s plain and simple,” Hart said. “I need to find a way to get back on track because I’m not playing well. I know it. Everyone knows it. It’s hard right now.”

Vigneault said he has coached many great goalies who have had to fight through slumps. He also said Hart needs to “battle a little harder.”

“I have to find a way to stop pucks. … and not overthink things and just trust my game because I’m just (expletive) right now,” Hart said in a somber tone. “I don’t feel like myself and I’ve got to find a way to get back into playing the game that I love.”

After James van Riemsdyk’s early goal, Buffalo scored three straight times in the first period as the Flyers’ defense was invisible, and Hart failed to make a clutch save when needed.

Sam Reinhart, who had two first-period goals, made it 1-1 as he beat Hart to the glove side from the left circle with 15:27 left in the first. That snapped the Flyers’ 185:44 shutout streak against Buffalo this season, the longest against any of their opponents in franchise history.

Riley Sheahan gave the Sabres a 2-1 lead by scoring on a backhanded rebound with 10:21 to go in the period. Seven minutes later, an unguarded Reinhart increased the advantage to 3-1 as he scored from deep inside the right circle.

Hart, who is 4-5 in his last nine decisions, was replaced by Elliott at the start of the second period. It was the third time Hart — who seems to be playing deeper in the net than usual and has been vulnerable on high shots to the glove side — was yanked in 15 starts this season.

“At the end of the day, it’s the toughest position in hockey,” Vigneault said. “Carter is a very young goaltender. We all believe in him. We all think he has a tremendous amount of potential. He’s going to learn from this and he’s going to get better. I believe he has to battle a little harder in goal at this time.”

The Flyers swarmed the net early in the second, and Scott Laughton, who missed a handful of first-period shifts after getting leveled by Curtis Lazar, set up Kevin Hayes for his ninth goal, getting Philly to within 3-2 with 17:02 left in the period.

» READ MORE: Flyers’ first game with fans was eerie; I locked down a year ago today, just before the sports world stopped | Marcus Hayes

Defenseman Brandon Mountour, scoring his first goal in 31 games, made it 4-2 when his left-circle shot from a bad angle deflected off Gostisbehere’s stick and over Elliott’s shoulder with 13:40 to go in the second.

Claude Giroux cut it to 4-3 as he converted a sweet behind-the-end-line feed from Voracek with 4:55 remaining in the second. Giroux, who has three goals in the last four games, got inside position on defenseman Henri Jokiharju on the doorstep and scored just after a Flyers’ power play had expired.

The Flyers (13-7-3) are in the midst of a key four-game homestand that started with Sunday’s 3-1 loss to Washington. They host the Capitals on Thursday and Saturday as they try to make up ground in the crowded East Division race.

“Its obviously important,” Hayes said before the game, referring to the homestand. “You look at the standings and everybody is right there and within a couple points. When you have home games, you have to take advantage.”

They took advantage of a scuffling 6-14-4 team Tuesday, but falling behind by two goals against the powerful Capitals (13-6-4) is not advisable.