In a statement, the FTSE 250-listed firm said Westenberger will leave on May 16, when the firm will also publish its half-year results.
The company said its board has already started a search for a permanent replacement for Westenberger, who had been finance director at the group for five years.
Brewin Dolphin also said its business "continues to perform in line with expectations, with continuing growth in income and profitability.”
In November last year, Brewin Dolphin reported a 15% rise in full-year first-quarter profit to £57.6mln.
The group’s assets under management rose by 3.5% in the quarter to end-December to £41.5bn, it said in January.