The job has been vacant since Edinburgh MP Ian Murray resigned, calling on Jeremy Corbyn to stand down as leader.
Mr Anderson was also made shadow Northern Ireland Secretary last week.
The north-east of England parliamentarian currently sits on the Scottish Affairs Committee at Westminster.
Mr Anderson said he would seek a meeting with Scottish Labour leader Kezia Dugdale "as soon as possible".
The Labour MP for Blaydon, Dave Anderson, is to be the party's new shadow Scottish Secretary, a senior source has told BBC Scotland.