Luol Deng and Andres Nocioni added 14 points each for the Bulls (2-4).
Rose, the first overall pick in the draft, went 13-for-18 from the field and had seven assists.
The Mavs built an 83-60 lead with 4:17 left in the third quarter on Erick Dampier's three-point play and Dallas carried an 89-72 edge into the final quarter.
But the Bulls rallied to narrow the deficit to 100-99 on Rose's two free throws with 1:38 left.
Aaron Gray's two foul shots with 1:09 remaining pulled Chicago even at 101, then Rose put the Bulls ahead for good 103-101 on a baseline jumper with 41.5 seconds left.
Thabo Sefolosha's two free throws with 29 seconds to play extended the Bulls' lead to 105-101, closing out the Mavericks.
Howard scored eight early points as the Mavs opened the game with a 17-5 run.
The Mavericks were in front after one quarter, 28-16, with Chicago going 4-for-21 from the field.
Dallas held a 59-43 halftime lead behind Nowitzki's 15 points and Howard's 12.
Ben Gordon, the Bulls' top scorer last season at 23 ppg., missed his fourth straight game with a jammed right big toe.
Chicago coach Vinny Del Negro said he'll be cautious with Gordon until the games start to count.
Michael Ruffin (sprained left ankle) has missed all six preseason games for Chicago.