O'Sullivan was outplayed by Joe Perry and trails 5-3 after the first session.
Perry raced into a 4-1 lead, but the reigning champion found his rhythm thereafter - not least in a break of 117 - before Perry pounced on an O'Sullivan mistake in the eighth frame to make a 99 break to clinch an overnight lead before the match resumes on Friday afternoon.
Australian Robertson - winner of the tournament in 2010 - raced into a 7-2 lead in the opening session against Williams, then rattled through the remaining three frames in the evening session with breaks of 61, 140 - the highest of the tournament so far - and 102.
"It is definitely the best I have ever played in the first round here," said Robertson, who went out in the opening round in 2011 and 2013.
"It was one of those matches that I was really determined to do well in to make amends for last year.
"It didn't matter who came out the hat, they were going to feel the wrath of some of my best snooker because this is the best World Championship I've ever prepared for in terms of the practice and how I feel leading into it.
"I had to come out and destroy him tonight."
Robertson, who has won the UK Championship and Wuxi Classic this season, faces Mark Allen in round two.
"It's the sort of snooker that's required to the win the World Championship this year and it's good to get it out of my system nice and early because I've got a really tough match with Mark Allen next.
"This is definitely the best season I've ever had in terms of the way I've played. I've won two ranking titles but with a fair reflection on how I've played, I probably should have at least four…
"Allen is someone you'd probably expect to meet in the quarter-finals, but I've got him in the last 16 so it's up there with the toughest draws you can get.
"We had a close one in the Masters when we both weren't at our best, but we've had a lot of really good matches - they always go quite close.
Dominic Dale won his first match at the Crucible in 14 years as he beat Mark Davis 10-5.
Dale, who came through qualifying, raced into a 9-2 lead then held his nerve despite a Davis fightback to claim the victory - his first since beating Davis Gray 13-1 in the second round back in 2000.
"Just being here is brilliant when you soak up the atmosphere," said world number 24 Dale.
"I have had some rock-hard draws here, I've had Judd twice and Ronnie in recent years.
"Obviously when my name came out against Mark Davis I thought I've beaten him the last couple of times I've played him so there's every chance this time.
"He's a great player and I'm a great player so I suppose it's who deals with the situation best on the day."
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