Basilashvili will face either Alexander Zverev or Taylor Fritz in the semi-finalStefanos Tsitsipas has been knocked out of the Indian Wells Masters after falling to a quarter-final defeat by unseeded Georgian Nikoloz Basilashvili.
Basilashvili capitalised on the out-of-sorts Greek world number three to win 6-4 2-6 6-4 in California.
The big-hitting 29-year-old had not won a main draw match at the tournament before this week.
Basilashvili will now face either Alexander Zverev or Taylor Fritz for a place in the final.
It is the first time world number 36 Basilashvili has reached the semi-finals of a Masters tournament.
French Open runner-up Tsitsipas, seeded second, started in sluggish fashion as he dropped the opening set, managing just eight winners to Basilashvili's 14.
However, Tsitsipas - who won his first Masters 1000 title at the Monte Carlo Masters in April - regrouped to dominate the second set.
But just when it looked like the 23-year-old was poised to replicate comeback wins over Fabio Fognini and Alex de Minaur in the previous rounds, Basilashvili secured a break in the opening game of the third and produced some superb winners to upset Tsitsipas.
