Kirk Hinrich added 18 points for the Bulls, who have won two of three to open the season. Luol Deng and Ben Gordon chipped in 11 points apiece in the win.
Darko Milicic and Mayo hit a basket each, before Gordon nailed a three, and Warrick had a dunk to pull Memphis within 78-76 with 10:53 left in regulation.
With the Bulls still up by two 3 1/2 minutes later, Gordon went 2-for-2 from the line, Rose had a dunk and Gooden drained a jumper -- countered only by a 1-of-2 performance by Milicic at the line -- that made it 87-80.
Following a timeout, Hinrich hit a free throw after a technical foul on Warrick. Gooden then blocked Mayo on a layup attempt at the other end and Lowry was called for a flagrant foul on Rose, who hit both from the line to make it 90-80 with 4:15 left to play.
The 10-point lead was Chicago's biggest to that point and lasted to the end of the game.
The teams played to a 19-19 tie after 12 minutes of basketball, but the Grizzlies took a 50-42 lead into the intermission.
The teams battled neck-and-neck in the third quarter. The lead went back-and- forth before a Warrick jumper and a pair from the charity stripe by Lowry tied it up at 66 with 2 1/2 minutes remaining prior to the fourth.
Moments later, Lowry again went 2-for-2 from the line to give the Grizzlies a 70-69 lead. However, Gordon connected on a pair of baskets that bookended a 2-for-2 performance by Aaron Gray, and Chicago went into the final stanza ahead 75-70.