In her first appearance in a WTA Tour main draw, world number 231 Branstine sealed a breakthrough 6-4, 5-7, 6-1 win over defending champion Samsonova.
Their match had been halted on Tuesday due to darkness at one set apiece, but Branstine returned to dominate the decider, not facing a single break point while converting two of her four break opportunities.
Branstine, who was born in California and represented the United States until 2017, will next face fellow qualifier Elena-Gabriela Ruse of Romania.
Fifth seed Hubert Hurkacz overcame a back issue, requiring two medical timeouts, to defeat Spain's Roberto Bautista Agut 7-6(2), 6-4 and book his spot in the second round.
The 28-year-old Pole will next take on 22-year-old Estonian Mark Lajal, who beat Serbia's Laslo Djere 6-2, 6-4.
"It's my first time here and I'm really loving it," said Hurkacz.
"It's always nice to play on grass. Obviously, the end was a little bit tricky, but hopefully I will be better for tomorrow. My back hurts a bit, but I will do some work with the physio later."
Khachanov, the third seed, prevailed in a third-set tie-break to claim a hard-fought 6-3, 5-7, 7-6(4) victory over American Mackenzie McDonald and reach the quarter-finals.
The Russian will play Canadian Gabriel Diallo, who saved two match points to overcome sixth seed Jordan Thompson 4-6, 6-1, 7-5.
Britain's Dan Evans came from a set down to defeat Australian Rinky Hijikata 3-6, 6-4, 6-3, and Greece's Maria Sakkari eased past Dutch wildcard Anouk Koevermans 6-3, 6-4.