Rublev dropped his first set of the week but ultimately prevailed with a 6-1 3-6 6-3 victory Basilashvili.
The Russian has won his past four finals at ATP 500 level and will seek a fifth on Monday (AEST).
"It's my first final on grass and in Halle," he said. "I think I can play on every surface and I will try my best again.
"I had good opportunities to break Basilashvili in the second set, some quite easy forehands and I stressed a little showing my emotions.
"I then came back and stayed calm, until the last game. But I won."
Rublev will take on unseeded Ugo Humbert, who held his nerve to edge a thriller against Felix Auger-Aliassime, winning 6-4 3-6 7-6 (7-5).
Humbert beat Alexander Zverev earlier in the week and has had to win a deciding set in all four rounds, while the beaten Auger-Aliassime had previously seen off Roger Federer as part of a dramatic event.
Frenchman Humbert won each of his first two career finals, which both took place last year in ATP 250 events.