NEW YORK — Ever since Rajon Rondo blew out his right knee, Jared Sullinger’s creaky back required surgery, and players such as Jason Terry and Courtney Lee failed to meet expectations, the margin for error for the Celtics in this postseason is razor thin, hardly any room for mistakes.

During the recent Big Three era, the Celtics strolled into the postseason realizing they could win their first-round series on pure talent, with better execution required in later rounds. But they did have an opening series to sharpen their proficiency.

That is not the case during this first-round series with the New York Knicks. The Celtics know they have to play near-perfect basketball to win the series because the Knicks are more dynamic offensively and have more depth. And the Celtics played well enough to compete into the fourth quarter Saturday, holding a precious 7-point lead late in the third.

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Their fourth-quarter collapse — zero points in the first 3:36 led to just 8 in the final period — was the reason for their 85-78 loss. But the Celtics hardly were discouraged by that outing and they shouldn’t be entering Tuesday night’s Game 2 at Madison Square Garden.

While the Celtics stumbled into the postseason, beating just one team with a winning record in the final six weeks, coach Doc Rivers appeared confident a different club would arrive in time for the postseason, one that would focus more on defense, rebounding, and ball security. The Celtics accomplished the first two goals in Game 1 but failed miserably at the third.

So they have approached practice the past two days with a confidence that wasn’t always prevalent during the regular season. They lack enough playoff-savvy veterans to simply rely on transforming into a rugged defensive team when the postseason kicks off, but for at least one game, they did.

They turned Game 1 into a halfcourt, possession-by-possession game, which is the way to play the up-tempo Knicks, who capitalize on opponents’ mistakes, especially Carmelo Anthony, who may be the best cherrypicking floater in the league during chaotic times on the court.

Anthony doesn’t dive for loose balls, he waits at the 3-point line for his teammates to emerge from the scrum and make a backbreaking long-range shot. The Celtics reduced those opportunities in Game 1 and will have to fare even better in Game 2.

The home teams were 8-0 in Game 1s over the weekend, but the Celtics and Golden State Warriors were the only teams that threatened to steal a win. The Celtics have taken on a different personality, understanding that playoff basketball is not run-and-gun or 3-point barrages.

“We played hard, and so did they, but we have to play better,” Rivers said. “Like I told our guys, hard is great. But hard and smart is more important. We made a lot of hard plays, even Courtney saves the ball inbounds [to New York’s Kenyon Martin for a layup]. You give him an ‘A’ for effort but the rest of the part is where you have to be smarter as a group. I thought we did a lot of that.”

The Celtics were guilty of making dumb mistakes — such as Lee saving the ball under the opponent’s basket or Avery Bradley making an entry pass from one side of the court to the other — and still they were a couple of made open baskets, primarily by Terry and Kevin Garnett, from stealing the opener.

For the past three years, the Celtics have been nearly allergic to the easy basket. And they have to find a way to score besides a Paul Pierce fadeaway, a Garnett 20-footer, or a Terry 3-pointer. While Jeff Green has used his athleticism to create layups over the second half of the season, they have to find a way to get the 7-foot Garnett near the basket against a hobbling Tyson Chandler, an undersized Martin, or an overwhelmed Chris Copeland.

Garnett not only has to be more assertive, his teammates have to offer better entry passes and also penetrate into the lane and feed the big man for a dunk. The Celtics have to make their own lives easier.

“We’ve just got to be a little more patient and take our time,” Pierce said. “Who’s to say that we’re going to hold two of their starters scoreless or J.R. [Smith] is going to shoot bad or Carmelo is going to have a huge night. Every game plays out differently but I think the constant thing is our effort on the defensive end and being able to control our turnovers.”

Game 2 won’t play out the same as Game 1, but the Celtics definitely gave themselves a self-esteem boost with their defensive performance, realizing they are capable of beating the second-seeded Knicks with a touch better execution. But then again, the Celtics lost a handful of games during the regular season with late collapses.

So Tuesday night could determine whether the Celtics are truly an upgraded and improved product over a maddening regular season. If they’re not, it likely will cost them the series.