The world number three Williams has been reminiscing about his early years in the sport during a lockdown chat on Instagram with the retired seven-times world champion Stephen Hendry.
Williams - world champion in 2000, 2003 and 2018 - recalls how he was handed a little piece of green baize folklore by the late Higgins during a match with Hendry at the UK Championship when he was only 17.
In his first year as a professional, Williams faced his friend Hendry in the last 64 of the event, their first encounter on the World Snooker tour. He was leading 6-5 when he bumped into the iconic and temperamental Northern Irishman - world champion in 1972 and 1982 - during a toilet break at Preston's Guild Hall.
"I remember when Alex Higgins gave me this little silver pig," Williams told Hendry.
"I think I was 6-5 up, and I went to the toilet. He's come in behind me and he's said: 'Mark, here is my little silver pig. It's been my lucky charm for however many years, I'd like you to have it'.
"I can't say the exact words, but he says: 'Do me a favour, go out and beat that &%£$"$"..any words you can think of..Stephen Hendry..I can't stand that $&$£$'.
"I'm trying to get past him because I'm thinking: 'I'm going to get docked a frame here, and I'm in front of you. He's massaging my shoulders and saying: 'Come on, don't lose the game.
"Out I've gone, and you've beat me in the end. I was cleaning up yesterday, and I found the pig again so I might put it on. That's 28 years ago. I don't know why he gave it to me, but he did and to be fair I always got on well with him.
"I didn't know him that well, but every time I was somewhere, he'd always come over and I'd have a chat with him."
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Williams led 8-6, but suffered a 9-8 defeat to the then world champion Hendry. Williams recalled he was given the celebratory fist by the exuberant Scotsman after losing the final frame.
Hendry admits his relationship with Higgins began to deteriorate when he started dominate the sport at the outset of the 1990s.
"I got on well with him in the beginning, but once you start winning you become the enemy," said Hendry.
"He didn't like you," agreed Williams.
Hendry was laughing when he recalled the unpredictable nature of Higgins, who tragically died in 2010 after years battling alcoholism.
"The last time I beat him, he said to me: 'up your a** you %£$£$ when we shook hands," said Hendry.
"I remember it was first to nine and I was maybe 6-2 up going into the evening session.
"There was eight tables at the UK Championship. I was waiting to play Alex and (six-times world champion) Steve Davis was waiting to play someone. Steve was standing there like a robot without moving.
"I'm standing there and Alex comes in. You are thinking: 'What's going to happen here'. He goes to Steve: 'I'm going to f*** him tonight Steve.
"Steve just moved his eyes sideways."