Clijsters crushed her Russian opponent Alla Kudryavtseva 6-2, 6-0 to advance to the third round.
Jankovic also enjoyed an easy win, defeating wildcard entrant Coco Vandeweghe, winning 6-2 6-1 in just 77 minutes.
Other top seeds to enjoy straight set wins were Francesca Schiavone and Vera Zvonareva.
Current French Open champion Schiavone beat Zuzana Ondraskova 6-2, 6-0. Third seed Zvonareva had a harder time against talented Timea Bacsinzky, eventually coming through in straigh sets, 6-3 7-6 after losing five straight games in the second set.
Marion Bartoli and Anastasia Pavlyuchenkova were also among those to move ahead, whilst 12th seed Petra Kvitova suffered a loss in 3 sets to Barbora Zahlavova Strycov 3-6 6-2 7-5.
