Five-time world champion O'Sullivan took the first three frames and won 4-1 after White had beaten Ryan Day without losing a frame.
Home favourite Higgins beat compatriot Scott Donaldson and fellow Scot Stephen Maguire defeated Fergal O'Brien.
Neil Robertson and Judd Trump were among Wednesday's other winners.
Robertson overcame Gary Wilson, while Marco Fu beat Liam Highfield and Trump defeated Igor Figueiredo, with Ali Carter losing to Chris Wakelin.
Scots Graeme Dott and Anthony McGill lost to Mark Davis and Mitchell Mann, respectively.
Mei Xi Wen eliminated Peter Ebdon and will play four-time world champion Higgins on Thursday.
Trump will face Noppon Saengkham and Robertson takes on Joe Swail.
The latest event in the Home Nations Series, the tournament is the first World Snooker ranking event held in Scotland since the World Open in 2010.
Ronnie O'Sullivan beat Adam Stefanow to set up a third-round tie against Jimmy White at the Scottish Open, as John Higgins also progressed in Glasgow.