Lucas stung Miami with 11 points in the fourth quarter, including two clinching free throws that set the final score with six seconds left.
"I'm a shooter and they're free throws so they're supposed to go in," Lucas told reporters after making 9 of 12 shots from the field. "Our captain went down. (It was) time to step up."
Rose sat out the game due to a groin injury but the Bulls still managed to open up a 3 1/2-game lead over the Heat in the Eastern Conference.
Dwyane Wade scored a game-high 36 points and LeBron James put up 35 but the Heat (31-11) fell behind by 17 in the third quarter and struggled to claw their way back.
Wade erupted for 19 points in the fourth quarter to pull Miami within four points in the final minute, but he missed an important three-pointer and the Bulls ended the game at the free throw line.
"Their second unit came in and provided a spark and took the game over," James said.
Chicago (36-9) led for virtually the entire game and outrebounded the visitors 50-34 to help them earn a measure of revenge for last year's loss in the Eastern Conference Finals.
The Bulls have endured injuries the entire season, but they saw the return of forward Luol Deng and point guard C.J. Watson, who started in place of Rose before giving way to Lucas.
Joakim Noah added 14 points for the Bulls.