First-half tries by South African forwards Hanro Liebenberg and Jasper Wiese when Saracens were down to 14 men swung the game Leicester's way as they led 12-6 at the break.
Leicester dominated possession in the second half but Saracens defended superbly and forced their way back, earning two penalties which Farrell slotted to level the match with four minutes remaining.
Leicester regrouped, however, inching their way within range for Burns, who had not scored a drop goal all season and was on as a first-half replacement for George Ford, to settle it.
