Djokovic breezed through the first set of his match against Italian Fabio Fognini but then faltered before regrouping for a 6-0, 5-7, 6-2 victory over his unseeded opponent.
Next on his agenda will be Bulgaria's Grigor Dimitrov, who notched straight-set 6-4, 6-4 decision over Australian Matthew Ebden.
Murray also had to rally in his second-round match Sunday, rebounding from a dropped first set to record a 5-7, 6-2, 6-2 victory over Russian Evgeny Donskoy.
He now advances to face Taiwan's Yen-Hsun Lu, who needed a pair of tiebreakers to get by Slovakian Martin Klizan 7-6 (7-5), 7-6 (7-5) in his second-round match.
Also Sunday, seventh-seeded Argentine Juan Martin Del Potro downed Nikolay Davydenko of Russia 6-3, 6-4; No. 8 Jo-Wilfred Tsonga of France bested American James Blake 7-6 (8-6) 6-4; 11th-seeded Nicolas Almagro of Spain defeated countryman Daniel Gimeno-Traver 7-5, 6-1; and 12th-seeded Croatian Marin Cilic bested Spain's Albert Ramos 7-6 (9-7), 6-2.
Other second-round winners included 19th-seeded German Tommy Haas; No. 16 Japanese Kei Nishikori; No. 23 Sam Querrey of the United States; 32nd-seeded American Mardy Fish; Germany's Bjorn Phau; Argentine Carlos Berlocq; and Australian Marinko Matosevic.