Butler made a 15-foot jumper at the buzzer to give Washington a 100-98 victory over the Toronto Raptors on Friday night.
Antawn Jamison had 24 points and 12 rebounds and Butler added 21 points for the Wizards, who nearly blew a six-point lead in the final minute.
Tapscott was overruled when he tried to substitute Butler in the fourth.
"You let the hot hand stay out there and let it stay as hot as it can," Tapscott said. "He played the entire fourth quarter. I was desperately trying to get him a rest. Every time I turned around to the assistants and said 'I think we should get Caron for a minute,' they said 'Let him play, let him play!'"
Butler wasn't interested in heading to the bench anyway.
"I was like 'I'm finishing this,'" Butler said.
He certainly did, sending defender Shawn Marion the wrong way with a pull-up drive, then making a jumper from the elbow as the buzzer sounded to give Washington the win.
"I was just trying to make sure there wasn't going to be any time on the clock," Butler said. "I was trying to size up the defender and Shawn Marion is a great defender. I was just trying to get him to lean one way and get the opportunity to get my jump shot off. Fortunately, it was able to happen."
Before the game, Toronto coach Jay Triano called Butler and Jamison "the best forward duo in the NBA as far as putting points up."
The pair started slow, shooting 3-for-13 in the first half, but combined for 36 second-half points as the Wizards won for just the third time in their past 13 games.
"I was flat, just flat," Butler said of his first-half struggles. "I was hitching the jump shot early on and I felt like I didn't have my legs underneath me."
A 3-pointer by Toronto's Jose Calderon cut Washington's lead to 88-87 with 3:30 left, but Jamison and Butler each made three-point plays on either side of a Chris Bosh free throw to give the Wizards a 94-88 edge with 1:41 remaining.
Washington led 96-90 with 47 seconds to go after a reverse layup by Juan Dixon but Jason Kapono made back-to-back 3-pointers from the corner, the second after a Washington turnover at midcourt, tying it at 96-all with 28 seconds remaining.
"It was scary," Butler said. "It was one of those situations where we felt we had the game under control. We made a couple of errors and they capitalized."
Dixon made a jumper and Calderon answered with a layup, giving the Wizards the ball with 16 seconds left and setting up Butler's game-winner.
"At the end, you want to go with the simplest thing you can do," Tapscott said. "You don't want to get too elaborate, you don't want to get too fancy, you don't want to outthink yourself or overcoach. Get the ball in the hands of one of your playmakers and let him make a play. That's what that was designed to do."
Bosh had 21 points and 10 rebounds for Toronto, and Kapono added 20. Calderon had 15 points and 11 assists.
Bosh was left shaking his head about Butler's buzzer-beater.
"It was a good shot, that's what he does," Bosh said. "We forced the jump shot, step-back jumper, and he made it. I think Shawn played good defense but he just made a tough shot."
"Brendan Haywood was terrific tonight," Tapscott said. "He really helped us shut down the lane a little bit."
The victory means Washington can do no worse than match the worst record in team history. The Wizards, who have two games remaining, went 19-63 in the 2000-2001 season.
"We really want to get that 20 wins and this was a great opportunity," Butler said. "We rode the momentum and got it."
Raptors center Andrea Bargnani, who missed Wednesday's loss at Indiana with a sore left heel, returned to the starting lineup and scored nine points.
Toronto never trailed in the first half, leading 20-16 after the first and 41-37 at the half, but couldn't hold on through the third.
Jamison had five points in the first half and Butler didn't score his first basket until a layup with 1:29 left in the second. But the pair came alive in the third, with Jamison scoring 11 and Butler adding eight as the Wizards took a 66-63 lead into the final quarter.
The Wizards won on the road for the first time since beating Minnesota 110-99 on March 9. ... Washington guard Gilbert Arenas dressed but missed his third straight game, resting his surgically repaired left knee. Tapscott said Arenas could play Monday when the Wizards host the Raptors. ... Former Raptors guard Alvin Williams, whose career was cut short by knee problems, watched from a courtside seat.