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.

The Public Administration Committee said it would seek written evidence and publish an interim report before the summer recess later this month.
Eurosceptic Tories fear the rules are being amended to allow the government to campaign openly to stay in the EU.
But ministers say it is needed to allow them to continue their work.
The committee, headed by Conservative MP Bernard Jenkin, said the purdah probe would form the first part of a wide-ranging inquiry into the EU Referendum Bill, the proposed law that will authorise a referendum on the UK's membership of the EU by the end of 2017.
Mr Jenkin said he and his colleagues wanted to know why the government was planning to partially "disapply" the existing rules on government announcements in the four weeks leading up to the referendum.
The inquiry will focus on the existing rules, as set out in the 2000 Political Parties and Referendum Act, the government's case for amending them, how ministers plan to go about it and the impact it will have on the impartiality of the civil service.
In a vote on the issue last month, 27 Conservative MPs rebelled against their party, urging ministers to reinstate the full purdah period although the government won the vote after Labour abstained.
The government has said the existing rules would potentially prevent ministers from attending EU meetings and making decisions with a European dimension. They have insisted they will address MPs' concerns about this and other matters - such as the funding available to different sides and the length of the campaign itself - as the bill makes its way through the Commons.
The committee will publish its report on 22 July.

The former private, who is in her 20s, has accepted a conditional caution for misconduct in a public office.
The woman has agreed to make a payment of Â£40 to an armed forces charity.
She was arrested at her Nottinghamshire home under the Metropolitan Police's Operation Elveden inquiry into alleged corrupt payments to public officials.
In a statement, the CPS outlined details of the offence but did not name the newspaper or the barracks.
CPS lawyer Gregor McGill said: ''The evidence in this case was considered very carefully and although there was sufficient evidence to prosecute this offence, when considering the public interest it was decided that a conditional caution was an appropriate course of action.
''In accepting a conditional caution, an individual accepts responsibility for the offending set out.''
The former soldier was arrested in September. She was the 87th person to be held as part of Operation Elveden, which was launched after the phone-hacking scandal at the News of the World newspaper.
The operation was established to investigate allegations of inappropriate payments to police but was widened in January 2012 to include other public officials. It is being overseen by the Independent Police Complaints Commission.
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.
[NEXT_CONCEPT]
Plans to relax so-called "purdah rules" on government announcements in the run-up to the EU referendum are to be the subject of a quickfire inquiry by MPs.
[NEXT_CONCEPT]
A former soldier who agreed to obtain information from an army barracks for a newspaper will not face a trial, the Crown Prosecution Service has said.