Heat talisman Butler produced a typically dynamic performance to steer Miami to a precious win, leading the scoring from 12-of-16 shooting while laying on 12 assists.
Cody Zeller provided offensive support with 20 points while Kevin Love and Max Strus added 18 points apiece, with Tyler Herro finishing with 15.
Miami improved to 41-37 and remain seventh in the Eastern Conference standings, one win behind the sixth-placed Brooklyn Nets as they chase an automatic playoff berth.
Heat coach Erik Spoelstra was pleased his team had been able to rediscover their offensive rhythm after only managing to score 92 points in each of their previous two games.
"They're a very good offensive team -- we knew we were going to have to put some points on the board," Spoelstra said.
"We knew we'd have to put some pressure on them offensively and during the course of the game we were able to get them on their heels, and Jimmy controlled the tenor and tempo of the game."
But while Miami celebrated, Dallas were left contemplating another painful loss that leaves them closer to the ignominy of missing the postseason one year after reaching the Western Conference finals.
The Mavs remain 11th in the Western Conference with a 37-41 record and are now one game behind the 10th placed Oklahoma City Thunder, who occupy the last of the play-in berths.
Dallas coach Jason Kidd pinpointed his team's shaky defense as the reason for the loss.
"It wasn't the offensive side of the ball, it was the defensive side," Kidd said.
"We were trying to get stops and then when we did we'd just throw it away. We've got to do a better job of keeping the ball."
Crucially, Oklahoma City own the head-to-head tie-breaker with Dallas, meaning that they will still be ahead of the Mavericks even if they draw level on wins.
Dallas, who have lost six of their last seven games, have just four regular season games left to force their way into the playoffs.
Luka Doncic led the Dallas charge in Miami on Saturday with 42 points, 10 rebounds and eight assists while Tim Hardaway Jr finished with 31 points. Kyrie Irving had 23 points but struggled to stamp his influence on the contest.
"Timmy (Hardaway), Ky (Irving) and Luka kept us in the ballgame," Kidd said. "We just didn't get any production from the bench tonight."
The writing was on the wall for Dallas after the first quarter, with Miami pouring in the points to lead 44-31 heading into the second.
Dallas regrouped in the second period but still trailed by 12 points at the break.
Although Dallas threatened to sneak back into the game late in the fourth quarter, Butler's poise at key moments ensured Miami always managed to keep the Mavs at arm's length as they closed out victory.
In Saturday's other NBA game, the New Orleans Pelicans maintained their playoff momentum with a 122-114 win over the Los Angeles Clippers.
The Pelicans are seventh in the Western Conference on 40-38, half a game behind the sixth-placed Clippers.
