Britz scored his second try after the break and although Leicester rallied to 23-20 through Scott Hamilton's try, a fourth Carter penalty sealed victory.
Scotland coach Frank Hadden brings back Chris Paterson at fly-half while Rory Lamont and Simon Webster return with Scott McLeod replacing Jim Hamilton.
Edinburgh were forced into two late changes, with Scott MacLeod replacing Jim Hamilton (knee) at lock, and John Houston coming onto the wing for Andrew Turnbull (illness).