Top seed Agnieszka Radwanska had already been eliminated from the WTA Premier grass-court event on the opening day, meaning just four of the eight seeds will now contest the second round.
Jankovic's defeat came in relatively comfortable fashion too, losing 6-3 6-3 to Madison Keys, who will now face defending champion Elena Vesnina.
After a first set in which she was only able to win only one of her five service games, Jankovic was broken just once in the second as she made a premature exit.
The Serbian second seed joined Azarenka, who was on the receiving end of a 6-4 3-6 5-7 result against Camila Giorgi earlier in the day.
While Giorgi upset the odds to oust Azarenka, Errani's day ended in disappointment as she suffered a 6-7 (5-7) 2-6 defeat at the hands of qualifier Lauren Davis.
There was further Italian disappointment as Roberta Vinci - Errani's doubles partner - and Francesca Schiavone lost to Daniela Hantuchova and Ekaterina Makarova respectively.
Flavia Pennetta gave the Italians something to cheer with a 6-3 6-3 win over Hsieh Su-wei.
There was double delight for the home fans at Eastbourne as British duo Heather Watson and Johanna Konta saw off Tsvetana Pironkova and Belinda Bencic.
Elsewhere, Varvara Lepchenko booked her place in the second round with a 5-7 6-4 6-1 win over Shuai Zhang.