Oleksy, 36, scored a stunning overhead kick for Warta Poznan in November against Stal Rzeszow, planting one of his crutches into the ground while turning mid-air to fire into the net.
His goal trumped Richarlison's acrobatic volley for Brazil against Serbia and Payet's sublime, long-range strike to win the award and become the first amputee to ever win the award. Goals from Paris Saint-Germain forward Kylian Mbappe and Manchester United striker Alessia Russo were nominated in the original shortlist before it was whittled down to a final three.
"It's been difficult for me to even dream about an award like this," Oleksy said. "I am standing before you today receiving the award for the best goal in the world.
"I am dedicating this award to Ewelina, Tomek, Tosia -- because it was thanks to them that I was able to pick myself up after the accident and am here now, and can score such beautiful goals.
"I also want to thank the entire football community because someone has created this beautiful sport that makes it possible for us to find fulfilment. Thank you very much!"
Oleksky later added at a news conference: "I looked at the crowd and thought: 'So you can compete with Mbappe? Yes you can! It is possible to win against him? It is!'"
Speaking to ESPN earlier this month, Puskas winner Oleksy said scoring his award-winning goal "felt like magic," adding that he did not expect it to receive worldwide plaudits.
"When I saw how the ball was played toward me, at the height it was, I thought 'this could be something beautiful,'" he said. "So I just hit it. I was following it with my eyes, and when I saw it in the net it just felt like magic.
"During practice, we don't really often try to do these kinds of shots. But when I started playing amputee football, I always wanted to do something special."
"I found out the next day, because the big TV stations and sports outlets started to show it. I knew then that it was going to be something big in Poland, but not across the whole world."
Oleksy, who made five appearances for Poland at the WAFF Amputee Football World Cup in Turkey last year, joins names such as Mohamed Salah and Son Heung-min on the list of Puskas winners.
"It was an honour to be nominated for Puskas Award. Congratulations @airoleksy, your goal is a masterpiece and you are an inspiration for all of us!," Richarlison wrote on Instagram, alongside his picture with Oleksy.
Oleksy combines football with a full-time job as an excavator operator.
"I invite Robert Lewandowski to a training session. I'll show him how to score nice goals," he joked.
He was awarded the honour on the night of FIFA The Best awards. Lionel Messi won Best FIFA Men's Player while Barcelona's Alexia Putellas won Best FIFA Women's Player.
Information from Reuters was used in this report,