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.
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"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
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