The Spanish world No. 3 looked in complete control early on, winning seven consecutive games to dominate the opening set.
However, Basilashvili fought back strongly to turn the match into a much tougher contest than Alcaraz had initially expected.
Alcaraz broke ahead in the second set but allowed the Georgian to recover after making several mistakes while serving at 3-2. Basilashvili raised his level and pushed the contest into a tie-break, where both players had opportunities to finish the match.
The Spaniard had to save three set points before eventually converting his fourth match point to complete the victory after one hour and 41 minutes. The result extended his winning streak at ATP 500 tournaments to 23 matches and earned him a 60th tour-level quarter-final appearance.
“I played great tennis and I’m really happy with the performance that I did today,” Alcaraz said. “I let him get into the match again. After that, I would say it was really close.”
Alcaraz admitted that the tie-break had been particularly challenging, with Basilashvili coming close to forcing a deciding set. However, the Spaniard was pleased with his ability to fight through the difficult moments and secure the win.
He will face seventh seed Tommy Paul in the quarter-finals after the American swept aside Luciano Darderi 6-1, 6-1. Alcaraz leads their head-to-head record 7-2 and defeated Paul in straight sets at both the Australian Open and US Open earlier this year.
Elsewhere, Daniil Medvedev strengthened his bid to qualify for the ATP Finals in Turin after battling past Cameron Norrie 4-6, 6-3, 7-6(1). The former world No. 1 saved a match point in the deciding set before dominating the tie-break.
Medvedev reached his 100th tour-level quarter-final and will meet qualifier Adrian Mannarino next. He currently sits fifth in the race for the ATP Finals, with the season-ending tournament scheduled for 15-22 November.
