The pair had traded frames to find themselves locked at 5-5 before Si won the next three to put himself on the verge of a stunning victory.
However, Williams won four of the next five to set up a final-frame decider. But Si held his nerve to claim a brilliant 77 break in that deciding frame to beat the three-time winner 10-9 and progress to the second round where he will face Wales' Jak Jones.
"I didn't play well all the way through the match, but I was playing well at the end," Williams said.
Williams paid credit to his 21-year-old opponent, who last year made it to the semi-finals of the tournament.
"That's one of the best breaks you will see under pressure and if he can do that more regularly then he is a future world champion," Williams said.
"He definitely has the potential to win the world championship. Whether it's this year, who knows, it will be tough."
Speaking about his own future at the Crucible, Williams added: "I'm 50 next year and I was looking around. I've loved every minute.
"I don't know if I will be back here again. I'm not considering my future but it's a tough game so to keep getting to these venues is hard so you have to treat it as if you're not going to get back.
"Fingers crossed you will see me playing next year but I'm 50 and can't keep going on forever."
Robert Milkins' match with Junxu Pang's also went the distance on Tuesday, with Milkins coming away with a tense 10-9 victory after both players had a chance to claim the win in the deciding frame.
Milkins will play David Gilbert in the next round after Gilbert knocked defending champion Luca Brecel out on the first day of play at the Crucible on Saturday.
