
Second seed Kuznetsova, one of eight Russians who began the week, crushed Czech Lucie Safarova 6-1, 6-2, firing 10 aces in less than an hour on court.
That rapid-fire win was in huge contrast to the three-hour marathon Chekvetadze needed to hold off Russia's unseeded Dinara Safina 6-7 (3-7), 6-4, 6-3.
Between them, the pair produced 17 double faults and just four aces in a see-saw affair which gave Chekvetadze more hope in her game after losing last week in her first Doha round.
Serb Jankovic, the number four, fought off frustration at her own game before ultimately controlling India's Sania Mirza 4-6, 6-3, 6-4
"I was fighting and struggling the whole match," said Jankovic, an Australian Open semi-finalist. "I got so angry at myself.
"It's difficult playing in your first match of the week, I had to get used to the conditions.
"She was by far the better player than I was, but I had some luck. I was frustrated and maybe tired - but I cannot make excuses."
Jankovic produced seven double-faults in the two-and-a-quarter- hours battle, losing serve five times while breaking the 32nd-ranked Indian six times. Each woman missed on 10 break chances.
Mirza is the darling of the Dubai crowds, with thousands of her compatriots based in the oil-rick emirate.
Chakvetadze and quarter-final opponent Jankovic have met with times with the Russian leading the series 6-2; she has won their last two meetings, both in 2007.
Kuznetsova, the 2004 runner-up, playing for the sixth consecutive year at the event, now faces doubles partner Amelie Mauresmo, who beat Japan's Akiko Morigami 4-6, 6-1, 6-4. "Im really pleased with how Im playing," Kuznetsova said after putting out of her mind a third-round loss against Sybille Bammer a week ago in Doha. "Im in focus and serving well.
"It took me a few days to get over that loss," she admitted. "I was unlucky, I led 4-0. I have to stay positive. Its important to take those losses in the right way."
The Russian whose best result this season was a runner-up in Sydney to Justine Henin, broke Safarova five times on her way to victory and fired ten aces.
By Bill Scott, dpa