Home-favourite Wawrinka was one of the favourites to clinch the ATP 500 event in Basle, yet the third seed fell to a 6-4 6-7 (1-7) 6-3 defeat at the hands of the Kazakh on Tuesday.
The 29-year-old was the architect of his own downfall with 55 unforced errors throughout the match, which was disrupted when high winds caused disruption to the power grid and caused problems for the arena's floodlights.
After recovering from losing the opener to level the match at one-set all, Wawrinka quickly found himself 3-0 adrift in the decider.
The Swiss recovered to level at 3-3, but crucially Kukushkin broke in the eighth game before serving out to record just his second win against a top-10 player in his career.
Milos Raonic is enjoying a career-best season and he remains in contention to make the ATP World Tour Finals next month after defeating American Steve Johnson 7-6 (7-4) 6-4.
The fourth seeded Canadian, who is making his debut at the tournament, hit down 12 aces but he feels he will have to play better as the tournament progresses.
"I really just stuck around in that first set," he said.
"Obviously I need to get significantly better for my next match, but I feel like I'm on the first track."
Croatian giant Ivo Karlovic - seeded eighth - too made it through to round two after beating Czech Lukas Rosol 7-6 (7-4) 6-3.
He now meets Kenny De Schepper in round two after the French qualifier successfully saw off Federico Delbonis of Argentina 7-6 (7-3) 7-5.
In other matches, Croatia's Ivan Dodig needed three sets to see off Swiss Marco Chiudinelli 6-4 6-7 (7-9) 6-4, while German Benjamin Becker eased past Portuguese Gastao Elias 6-3 6-2.
And Canadian Vasek Pospisil accounted for Finland's Jarkko Nieminen 6-3 6-7 (4-7) 6-4 in just under two hours.