Gasquet started the week 50 ranking points behind Jo-Wilfried Tsonga for the final spot in the eight-player tour finals and needs to advance deep into the tournament in Moscow to keep his chances of catching Tsonga alive.
He took a step in the right direction Thursday with a 6-3, 3-6, 6-1 win over Evgeny Donskoy. Gasquet dropped his serve twice Thursday, with both coming in the second set. He responded by winning the first five games of the third set to take control.
Qualifier Mikhail Kukushkin extended his run in Moscow with a 6-4, 6-4 upset of fourth-seeded Alexandr Dolgopolov. He won two-thirds of the points on serve and saved three of Dolgopolv's four break chances.
Russian wild-card entrant Teymuraz Gabashvili had dominant performances in the first and third sets of a 6-1, 6-7 (7-9), 6-1 decision over Dudi Sela.
Friday's quarterfinals send Gasquet against Gabashvili while Karlovic is matched against Karen Khachanov, another Russian wild card. Second-seeded Andreas Seppi meets Edouard Roger-Vasselin and Kukushkin faces Andrey Golubev in the other quarterfinals.