Clayton Donaldson should have put his side ahead when he skipped around David Button in the Brentford goal, only to fire disappointingly wide from a tight angle.
Tarkowski then flashed a header narrowly wide from a corner, before Donaldson's header from Paul Caddis's cross was beaten away by Button.
It was Caddis and Donaldson who combined for the opening goal when the Blues captain's cross was met by Donaldson, whose header deflected off Tarkowski and into the back of the Brentford net.
Donaldson was inches away from connecting with David Cotterill's dangerous cross shortly after the restart, before the Blues striker saw his appeals for a penalty waved away after a skirmish in the Brentford box.
Jon Toral flashed a header narrowly over as Brentford went in search of an equaliser, with Darren Randolph then keeping out Toral's low strike.