The 28-year-old Japanese is making a late charge to the eight-player ATP Finals, in what would be his first appearance in two years when it opens in London on November 11.
After Nishikori held a serve in a tightly contested first game, two Kukushkin returns into the net and a return ace saw Nishikori break in a love game.
After pushing for the break in the decider, Anderson finally converted on his sixth break point opportunity of the set to seal the win in the final game.
Huge comeback sends Stephens to WTA Finals title match vs Svitolina
I just going and kept playing and tried my absolute hardest, and I got my opportunity at 4-3 when I broke there. Tennis players spent on the game for two hours and 42 minutes, playing all the sets.
His qualification means there are two places still up for grabs ahead of next week's Paris Masters, with Croatian Marin Cilic, Austrian Dominic Thiem, Nishikori and American John Isner still in the race. Thiem now holds the eighth spot, which is the final place in November's season-ending tournament.
Nishikori is still in the running for a year-end championships spot, and lost his ninth consecutive final since winning at Memphis in 2016. Anderson added 19 aces to his tournament-record tally (56) and won 89% of 2nd-serve points won.
Anderson, who also won in NY in February and reached the final at Wimbledon, earned a fifth career title but first at a higher-level ATP 500 event.
Copil upset expectations of a Federer final against Alexander Zverev in a 6-3, 6-7 (6-8), 6-4 win over the fifth-ranked German in the earlier semifinal.
Anderson was runner-up to Novak Djokovic at Wimbledon this year but the win in Vienna represented the biggest title of his career.