Djokovic, who officially took over the world No. 1 ranking Monday, defeated seventh-seeded Jo-Wilfried Tsonga 7-6 (7-4), 6-3. Murray, seeded No. 3, was a 3-6, 6-3, 6-4 winner over fifth-seeded Tomas Berdych.
Djokovic and Murray are each 1-0 in Group A play; Berdych and Tsonga are 0-1. The top two players in each group after round-robin play advance to Saturday's semifinals.
Djokovic improved to 5-0 this year against Tsonga by winning four of the first five points in the tiebreaker to take the first set and recording the only two breaks of the match in the second.
Djokovic opened the second set with a break and closed the match with his other.
Murray battled Berdych for 2 hours, 16 minutes before finishing off the win. He lost only six points on serve in the first set, but Berdych, who fought off seven break points in the set, grouped four of them in the sixth game for a break. Murray evened the match with a break in the fourth game of the second set.
The third set again turned on one break as Murray took the advantage in the third game and dropped one point on serve the rest of the way for the win.
Group B play is to begin Tuesday, with two-time defending champion Roger Federer, now second in the world rankings, going against eighth-seeded Janko Tipsarevic and fourth-seeded David Ferrer playing Juan Martin del Potro, the sixth seed.
18-year-old elf alleges mall Santa pinched her buttocks on the job
Kate Moss Playboy shoot is classic Playboy, classic Kate