Playing in his 137th career postseason game, Pierce connected on four of his five shot attempts in the fourth as the Nets kept the Raptors at bay.
"You just get that feeling, you've been in those situations a number of times," Pierce said. "I don't get rattled in the fourth quarter, down the stretch or in playoff settings. I've been in pretty much every playoff setting that you can imagine. I just try to stay calm and bring my calmness to the game."
Nets coach Jason Kidd said Williams "set the tone" by scoring 18 points in the first half.
"He came out with high energy," Kidd said. "I thought he was looking to be aggressive in scoring the ball. He got the guys off to a good start."
"We didn't shoot well from 3 tonight but we made up for that by taking care of the ball, good defense down the stretch, making plays offensively," Williams said. "That's why we got the win."
The Nets turned the ball over nine times, while the Raptors had 19, leading to 17 Brooklyn points.
"That was the biggest issue," Toronto coach Dwane Casey said.
Valanciunas is the second Raptors player to have a double-double in his postseason debut. Tracy McGrady had 25 points and 10 rebounds against New York in his first playoff game in 2000.
"He grew up today and that was huge for us," Casey said.
Valanciunas also set a Raptors record for playoff rebounds by surpassing Keon Clark, who had 16 against Detroit in 2002.
"DeMar didn't have the best of games" Lowry said. "He'll bounce back for sure."
The game was delayed midway through the third quarter when the shot clocks above each basket malfunctioned. Play eventually resumed with both clocks still dark, and the stadium announcer marking the remaining time at 10 seconds, then counting down from five before saying 'Horn' as time expired. The clocks remained inactive for the rest of the game.
"It was definitely tough because you're used to looking up to see the time," DeRozan said. "We just tried to help each other out when the announcer called down from 10 seconds."
In a written statement, Raptors ownership blamed the outage on a "signal path failure" that also affected the backup clocks. The team said new cables will be installed and inspected by the league before Game 2 on Tuesday night.
Vasquez called Tuesday's game "a must-win" for the Raptors.
Trailing by five to start the final quarter, Toronto tied it at 67-67 with 9:50 left thanks to a 3 and a fast break layup from Patrick Patterson.
Brooklyn moved back in front thanks to four points from Livingston and a banked jumper by Mirza Teletovic, but the Raptors tied it at 73-73 with 6:25 left on Lowry's driving layup.
Toronto led 77-76 on a 3 by Vasquez at 5:15, but the Nets replied with a jumper by Johnson, a turnaround jumper by Garnett and a 3 by Pierce to lead 82-76 with 2:58 remaining. Garnett's basket was his first made field goal in five tries.
Kidd said Pierce's assist on Garnett's basket was indicative of Brooklyn's collective trust.
"That just symbolized who we are as a team, that we trust one another no matter what someone is shooting from the floor, how someone is playing, that we all trust that someone is going to step up and make a shot," Kidd said.
Pierce added two more baskets on either side of a 3 by Vasquez, and the Nets sealed it with six free throws in the final 22 seconds.
Kidd was asked before the game to explain why he had shaved his beard. "I thought it would be a nice thing to do to start the playoffs," Kidd said. ... Raptors GM Masai Ujiri used a profanity about Brooklyn in a pregame address to fans watching outside the arena. Ujiri said he was "just trying to get our fans going." ... Rapper Drake and former Raptors guard Alvin Williams attended the game.