Islanders center Casey Cizikas, third from left, celebrates his goal...

Islanders center Casey Cizikas, third from left, celebrates his goal with teammates during the first period of an NHL game against the Red Wings on Thursday in Detroit. Credit: AP/Carlos Osorio

DETROIT — It was a good, hard-fought win.

It was a solid sweep of a two-game road trip.

And it could signify that the Islanders indeed will mount a strong playoff push.

“It was such a big game for us,” Mathew Barzal said. “We knew it, too.”

Barzal banked in the winner from behind the goal line at 13:58 of the third period and Pierre Engvall clinched it with the Islanders’ first empty-net goal of the season in a 5-3 victory that snapped the Red Wings’ winning streak at six games on Thursday night at Little Caesars Arena.

The Islanders twice lost leads, including a two-goal edge in the first period, but notched consecutive wins for only the second time since Dec. 13.

“It’s a good road trip,” coach Patrick Roy said. “Dallas is not an easy place to play in [for Monday’s 3-2 overtime win]. Here, the same thing. They’re a fast team, they move the puck well. I think we can build on that.”

The Islanders (25-20-14), who got two goals from Brock Nelson and 23 saves from a brilliant Ilya Sorokin, moved within five points of the third-place Flyers, who have played one more game, for the Metropolitan Division’s last guaranteed playoff spot.

The Islanders are six points behind the Lightning, who have played three extra games, for the Eastern Conference’s second wild-card spot.

The Islanders have won four of their last five road games but go into Saturday night’s game against the Bruins at UBS Arena having lost four straight home games.

“Let’s give some nice hockey to our fans,” Roy said. “Let’s give our fans some wins. I think they deserve it. They’ve been very patient with us.”

The Islanders went 1-for-3 on the power play and killed off the Red Wings’ two man advantages, both in the first period.

Defenseman Olli Maatta’s one-timer from the left circle brought the Red Wings within 2-1 at 5:04 of the second period. Patrick Kane, skating the way he did in his Chicago prime, tied it at 2-2 just 10 seconds into the third period as he kept the puck on a two-on-one rush.

“It’s a good idea of what a playoff game looks like,” Red Wings right wing David Perron said.

But Nelson, with a power-play goal from the right circle, regained a 3-2 lead at 5:04.

“We knew this was going to be a tough game,” said Nelson, who opened the scoring after creating a turnover deep in the Red Wings’ zone at 12:11 of the first period. “We got a huge power-play goal but we let the lead slip, but then I thought it was a good response and a big goal by Barzy.”

Maatta’s second goal made it 3-3 at 10:49, but Barzal swiped the puck toward the net from the right corner and goalie Alex Lyon (22 saves) kicked it in with his back leg.

While top-liners Nelson and Barzal totaled three goals, the Islanders also got two goals from their revamped third line of Casey Cizikas between Engvall and Simon Holmstrom.

“I call that my Swedish connection,” Roy said of Engvall and Holmstrom.

Cizikas tipped defenseman Adam Pelech’s shot from the left point for a 2-0 lead at 16:18 of the first period.

“Offensively, I thought we were jumping quick on pucks,” Roy said. “Defensively on the breakout, we were playing south-north. Tracking was really good. The line of Cizikas with Engvall and Holmstrom had a solid night. In the third, I thought the line of [Bo] Horvat with Barzal and Nelson, they started to connect more. They started working more together like the other three lines.”

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