Bourque opened the scoring with a back-hander from 10 feet out that just slipped under Mike Richter's shoulder at 2:01..
"I haven't scored many from that position," said Bourque, who has 393 goals and 1,107 assists in 1,509 games all with the Bruins. "Most of my goals are from out further. I was more excited about the execution of the goal, and that it gave us a lead."
Even coach Pat Burns was impressed.
"That was amazing," Burns said of the way Bourque scored.
"I was a little screened on the shot," the Rangers goalie said. "Plus, he put it in a perfect spot."
Hlavac, whose wife Marika will undergo induced labor on Saturday with the couple's first child, put the Rangers ahead to stay at 17:17 on a power play when Bourque made an uncharacteristic unforced turnover in his own zone.
Petr Nedved, Hlavac's linemate, wound up with the puck ad fired a shot off the right post. The rebound bounced to Hlavac at the left post, and he tucked it in.
Defenseman Kim Johnsson scored 80 seconds later on a wrist shot from the left circle, and the Rangers took a 3-1 lead.
"It seems that every first period mistake we made, the puck was in the net," Burns said.
Rangers coach John Muckler said his team's puck movement made the difference.
"In the first period, we moved the puck better than we have all season," Muckler said. "We're a much better team when we're playing with flow than playing in our own end."
Boston forward Andre Savage narrowed the Rangers' lead with the only goal of the second period, a slap shot that knuckleballed past Richter at 16:25.
Hlavac scored twice in a 50-second span of the final period to seal the win and secure his first NHL hat trick.
Linemate Radek Dvorak stickhandled his way into Boston's zone and was pulled down. With the delayed penalty in effect, Dvorak made a pass on his stomach to Hlavac, who sent a one-timer past Dafoe at 9:22.
At 10:12, Hlavac took the rebound of his own shot on the left side and flipped the puck from a sharp angle into the top right corner.
After the game, most of the promotional caps the Rangers gave away before the game were piled in front of Hlavac's locker.
"I'm going to give a couple of those to my doorman," Hlavac said. "This night was the best night ever this year. Muck said I have to be shooting more. I'm shooting more."
The win also gave the Rangers a four-point lead on Boston as the teams fight for eighth place in the Eastern Conference.
"There's not a night we can now say that losing two points doesn't matter," Richter said. "Every game is like a playoff game now."
Bourque wished his milestone came in a win, but was still proud of reaching a mark only eight other players have attained.
"I never thought I'd get to that point," Bourque said. "It was the wildest dream I ever thought. It's nice, and now the 1,500 is down, and it's time to look forward."
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