England's Smith succumbed in a 10-leg tie-break on Thursday night to fall in the second round at Alexandra Palace in London.
A see-sawing final set was edged 6-4 by Doets, who averaged 105 in the decider to gain revenge on Smith after losing at the same stage last year.
Doets, the world number 53, said the feeling of victory was indescribable and he can "sense an opportunity" to progress further.
"Darts-wise, this is the best I have ever felt in my life," the Dutchman said.
"When I was 2-1 down, I felt a focus that I have never experienced before, even though I was so nervous.
"I can't describe this feeling. It's absolutely incredible, and I do sense a real opportunity now.
"My confidence is there, my game is there, and to get over the line gives me a lot of hope for the next game."
Doets' confidence will be tested against one of Krzysztof Ratajski, Richard Veenstra or Alexis Toylo in round three after Christmas.