The bank called a review in November last year as it looked to shift its marketing activity from above-the-line to digital.
Y&R did not repitch for the business and the review was handled by AAR.
Anomaly’s remit includes a "specific focus on driving innovation" across all Bank of Scotland communications.
Y&R, which was known as Rainey Kelly Campbell Roalfe/Y&R until earlier this year, was awarded the brand’s £8m ad account without a pitch in 2009, having worked with Lloyds Banking Group since 2002.
Lloyds Bank, the flagship Lloyds Banking Group brand, ended its 13-year relationship with Y&R in 2015 after awarding its creative account to Adam & Eve/DDB.
Gareth Haggerty, head of Bank of Scotland innovation comms, said: "We’re delighted to be working with Anomaly. They will help inspire bold, innovative and impactful work reflecting the fast-evolving Scottish market."