The world number 12 looked in control at 3-1 after some strong early pots but White came back in the fifth to make it 3-2.
However, O'Sullivan kept his cool to secure the final frame and set up a next round match with Mark Allen.
O'Sullivan, who was using a new tip for the game, told Eurosport after the match that he felt comfortable with it despite plenty of speculation.
Robertson sets up Cao final with win over Higgins
"I'm so happy with how I approached the match mentally," O'Sullivan told Eurosport.
"We both struggled a bit, but it's so easy to give up sometimes when you're playing like that, but I carried on enjoying it.
"I thought 'keep playing, keep potting balls, if you miss don't worry about it. Just keep playing until the end'.
"I enjoyed that match a lot more than when I played Jimmy in Ireland because he played with a bit more confidence, he'd won a few matches.
"I know I could have scored more and done whatever, but my mental skills, I was over the moon with."
Allen overcame Li Hang 4-3 in his third-round match and is scheduled to take on O'Sullivan at 8pm for a place in the last eight.
O'Sullivan, who was the runner-up in the English Open to Mark Selby, has been in good form so far in the Scottish Open and he had eased through his opening two games.
The third-round draw placed him against White, a good friend, and a player with whom he has had many battles with over the years.
There were also a few familiar faces in the crowd, with some of the Rangers players taking in the snooker before their game against Hamilton on Friday evening.
Scottish Open third-round results
Dominic Dale 4-3 Ricky Walden
Neil Robertson 4-2 Joe Swail
John Higgins 4-0 Mei Xiwen
Mark Allen 4-3 Li Hang
Mark Davis 4-1 Xiao Guidong
Ronnie O'Sullivan 4 - 2 Jimmy White
Mark J Williams 4 - 3 Stephen Maguire
Yu Delu 4 - 3 Daniel Wells
Marco Fu 4 - 0 Robin Hull
Sean O'Sullivan 4 - 1 Mitchell Mann
Robert Milkins 4 - 1 Joe Perry
Barry Hawkins 4 - 1 Mike Dunn
Anthony Hamilton 4 - 0 Chris Wakelin
Liang Wenbo 4 - 0 Ian Burns
Kyren Wilson 4 - 0 Oliver Lines
Judd Trump 4 - 0 Noppon Saengkham
Fu beats Higgins to win Scottish Open after stunning recovery
Higgins stuns Trump with astonishing recovery in semi-final thriller
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