Trump had looked in a strong position going into the final session as he held a 9-7 lead, but 31-year-old Vafaei, who came through two qualifying matches and was the only qualifier to win in the first round, fought back well.
He won six of the nine frames in the final session, making breaks of 106 and 91 to come from 12-11 down to win 13-12.
Vafaei, the current world number 32, felt he could go all the way in Sheffield, the city he lives in.
“I’m looking forward to playing anyone – I’m not scared of anyone and all of them know that,” said Vafaei, who plays China’s Wu Yize next.
“There’s no pressure, so why should I be worried? Some of us are different. When you see the line you get hungry or you get tired, it’s up to you. I came to live in Sheffield to be beside the Crucible. The Crucible trophy is living in my head rent free.
“It’s one of the biggest wins of my career. I’m looking forward to it against Wu and fancy the job – let’s see how talented he is. I was not nervous, I enjoyed every moment. I was born for these kinds of situations.”
Trump, the 2019 champion, added: “It hurts but I had my chances. When you know you didn’t play a good shot it’s not like you’ve been unlucky.
“I’m not angry about it, I just was not good enough.”
