WASHINGTON — The new backcourt worked.

Knicks coach Jeff Hornacek changed his starting point guard, removing Emmanuel Mudiay for Trey Burke, and it paid off in a rare 101-97 victory over the Wizards at Verizon Center.

Hornacek made a full changing of the guards as Burke, in his first start as a Knick, and rookie Frank Ntilikina started over Mudiay and Courtney Lee on Sunday.

With Burke converting a gigantic three-point play in the final 40 seconds and finishing with 19 points, the Knicks eked out the win against a playoff-bound club, building a 12-point second-half lead.

“I tried not to be emotional,’’ Burke said. “The beginning of the game, a few plays out of character, trying to be extra aggressive against my old team, coming back here, trying to be extra aggressive. Playing with these guys all last year, having a relationship with them, you want to do good and win the game.”

And so the Knicks did, but not without a big boost from, ironically, Mudiay, who had made 14 straight starts before Sunday. He looked much better coming off the bench, and helped juice the Knicks during a 42-32 second quarter. In perhaps his best game since his first as a Knick — he was obtained Feb. 8 — Mudiay posted 11 points, seven assists and six rebounds in 21 minutes during which he was a plus-8.

“I just approached it how I wanted to approach it,’’ Mudiay said. “I wasn’t approaching it in a bad way. You still got to play. It was a fun game.’’

Trailing by 1 with 35 seconds left, Burke snaked into the lane, made a runner and was fouled by Kelly Oubre. He hit the free throw to give the Knicks the lead for good at 97-95.

“I didn’t see no help-side man,’’ said Burke, who shot 8-of-15 with three assists. “That’s the only reason I attacked. I wanted to get a good shot. That was the best shot. All I had to do was beat Kelly. I knew he had to foul me or I’d have a wide-open layup. That’s my growth to reading the game, seeing where the defenders are, attacking defense when they’re not in the right areas.’’

Ntilikina made his fourth start this season, but all have come off the ball at shooting guard. He didn’t finish the game, though, as his offense lagged in a 1-of-6, two-point evening in 21 minutes. However, he helped bother All-Star Bradley Beal into a 5-for-17 outing.

Hornacek played small in the final minute, but Washington center Marcin Gortat blew a tip-in with 17.7 seconds left and Burke was fouled. He made one of two free throws to keep Washington alive. But Lee cinched it from the free-throw line with five seconds left.

Mudiay had been benched the final 43 minutes of Friday’s loss to Minnesota. Burke and Ntilikina started the second half for Mudiay and Lee on Friday and were much more efficient.

Hornacek admitted the team may have thrown Mudiay into the fire too soon and acknowledged management was part of the decision-making process. Sunday, Mudiay looked like all he can be — with a few quick, deep passes and one sensational head fake and burst to the hoop for a layup.

“We all knew we wanted to look at [him],” Hornacek said. “That’s why we traded for him. Sometimes it’s good. Maybe after the fact we put him in a tough situation. Some guys, put him in there and get right to it.”

Hornacek said defense is his biggest concern with Mudiay’s opening month as a Knick.

“One thing we wanted to get on him about was to pick up the defense,’’ Hornacek said. “The league is full of great point guards.”

Last season, Burke lost his spot in the Washington rotation after February. Revenge was sweet as the Wizards (40-33) jostle for playoff position.

“I was a little nervous, I ain’t going to lie,’’ Burke said. “It’s been a while. I haven’t started since the beginning of my second year. After the fourth minute I was relaxed. It feels good being back in the starting lineup running the franchise.”