The second-seeded Kvitova of Czechoslovakia seemed headed toward an easy victory with a 4-2 lead in the second set before the Serb began a dramatic rally.
"Playing on grass is very difficult for me; it does not come naturally," said the 30-year-old Jankovic, whose best finish in a Grand Slam event was runner-up to Serena Williams in the 2008 U.S. Open. "In the last couple of years I've played so poorly. Being on Centre Court against the defending champion was unbelievable and the crowd was great, so I'm really, really happy I was able to win this match.
"She was playing unbelievably and she's in great form, there's no doubt about that. I just tried to stay one point at a time, just hanging in there, stay positive and fight. When I won the second set I knew I had it, and I just had to keep going."
Jankovic broke back to even the second set before breaking Kvitova in the 12th game, when the defending champion drove a forehand into the net.
The deciding set was on serve until the 10th game, when the Czech's forehand deserted her at 30-love. She sent one into the net, hit another long and wide, and sent a third several feet beyond the baseline, and Jankovic had her first match point.
When Kvitova hit into the net with a backhand, Jankovic flopped to the ground, finding her victory hard to believe.
"From the beginning of the match I felt in a good way," said Kvitova, who also won at Wimbledon in 2011. "Suddenly I was just missing it. It was unusual, or weird, but it is how it is. She was going for it and playing more aggressively. I was still waiting for chances that didn't come."
Jankovic, who has never advanced past the quarterfinals, will face No. 13-seeded Agnieszka Radwanska of Poland on Monday.
In other women's matches, No. 5 Caroline Wozniacki defeated No. 31 Camila Giorgi, 6-2, 6-2; No. 20 Garbine Muguruza beat No. 10 Angelique Kerber, 7-6 (12), 1-6, 6-2; No. 13 Agnieszka Radwanska ousted Casey Dellacqua, 6-1, 6-4; No. 13 Agnieszka Radwanska beat Casey Dellacqua, 6-1, 6-4; No. 15 Timea Bacsinszky eliminated No. 18 Sabine Lisicki, 6-3, 6-2, and No. 21 Madison Keys beat Tatjana Maria, 6-4, 6-4.
On the men's side, No. 2 Roger Federer got past Sam Groth, 6-4, 6-4, 6-7 (5), 6-2; No. 3 Andy Murray took care of No. 25 Andreas Seppi, 6-2, 6-2, 1-6, 6-1; No. 6 Tomas Berdych defeated. Pablo Andujar, 4-6, 6-0, 6-3, 7-6 (3); No. 9 Marin Cilic ousted No. 17 John Isner, 7-6 (4), 6-7 (6), 6-4, 6-7 (4), 12-10; No. 22 Viktor Troicki eliminated Dustin Brown, 6-4, 7-6 (3), 4-6, 6-3; No. 23 Ivo Karlovic defeated No. 13 Jo-Wilfried Tsonga, 7-6 (3), 4-6, 7-6 (2), 7-6 (9); and No. 20 Roberto Bautista beat Nikoloz Basilashvili, 7-6 (4), 6-0, 6-1.