Jankovic put them ahead with a 6-4, 6-4 victory over world number 157 Hsieh Su-wei but Djokovic was pushed hard before beating 110th-ranked Lu Yen-hsun 7-5, 2-6, 6-3.
Having only begun practising two weeks ago after undergoing nose surgery in November, world number three Jankovic looked very rusty as she struggled for consistency.
The 21-year-old Hsieh, who has played most of her matches on the ITF circuit, a level below the main WTA Tour, forced Jankovic into a host of errors but the Serb's greater experience saw her through.
"It was my first match of the year, after two months off (and I had surgery), I didn't expect myself to play my best tennis but I really enjoyed myself and am happy with the win," Jankovic told reporters.
Men's world number three Djokovic needed to dig deep to overcame a stubborn effort from Lu, who played superbly to level the match at one set apiece after the Serb had won a tight opener.
Djokovic had treatment on his lower back early in the second set but he recovered his poise in the decider to clinch a battling victory.
France opened their bid for the title with a 2-1 win over Argentina in the day's other Group A match. France owed their win over Argentina to a superb mixed doubles performance as Arnaud Clement and Tatiana Golovin beat Juan Ignacio Chela and Gisela Dulko 6-3, 6-2.
Golovin wore down Dulko 6-4, 6-3 to put France ahead. Chela, however, put Argentina level with a 6-3, 6-1 victory over Clement.