The No.3-seeded Radwanska was the first to take the court, but she spent far longer on there, needing exactly two and a half hours to finally put No.27 seed Sorana Cirstea away in three, 67(3) 63 64.
Radwanska could have won the first set - she held two set points leading 5-4 before losing it on a tie-break - but she went right to work again from there, winning 11 of the next 15 games to take the second set and build a 5-1 lead in the third. Cirstea closed to 5-4, but lost serve to lose the match.
"She was playing very good tennis from the beginning to the end," Radwanska said. "Every set was very tight, and I was getting a bit nervous at 5-1 in the third set when she started to play unbelievable tennis - she was hitting the ball so well. But the most important thing is I won the match in the end."
Awaiting Radwanska in the fourth round will be No.13 seed Maria Kirilenko, who spent even longer on the court - two hours and 44 minutes - to make it past American qualifier Mallory Burdette, 63 46 62.
Radwanska leads Kirilenko in their head-to-head, 6-2, and has won their last five meetings - but their last two meetings both went to 7-5 in the third set, at Indian Wells in 2011 and Wimbledon in 2012.
The No.5-seeded Kvitova had some late trouble but beat Ukrainian qualifier Lesia Tsurenko, 62 76(5).
"I started pretty well today - I knew what I had to do, and I didn't make a lot of mistakes," Kvitova said. "It was a nice first set for me, but then she came back and started playing more aggressively, and she started making me make some more mistakes. She came back very strong in that second set.
"I just had to stick with my game, focus on holding my serve and try my best on every point."
The fourth round is Kvitova's best ever result at Indian Wells. "It's a nice job to get my best result here," she said. "The surface doesn't suit me very well - it's quite slow - but I'm trying, and I hope I can just keep improving my tennis and my standard will go higher. I'm still working hard on everything."
This is also Kvitova's seventh straight win since falling to Serena Williams in the quarterfinals of Doha a few weeks ago - a match where she led 4-1 in the third set before getting edged, 36 63 75.
"I got a lot of confidence from that match against Serena," said Kvitova, went on to win Dubai the next week. "It showed I can still play with and beat the top players in the world. I need these matches."
Zakopalova has beaten Kvitova in both of their previous meetings, but there's one asterisk there - those two meetings both took place all the way back in 2007 at ITF Women's Circuit tournaments.