Having bowled out Bangladesh for 89 on the opening day, Sri Lanka went to lunch on the second day at 352-5, an overall lead of 263 runs with five wickets in hand.
Lanky left-hander Vandort, a late replacement for the injured Upul Tharanga, hit 117 with the help of nine boundaries and three sixes.
Jayawardene, who had retired hurt the previous day on 93 due to leg cramps, returned at the fall of Vandort's wicket and duly completed his 17th Test century.
The captain was unbeaten on 126 at the break with wicket-keeper Prasanna Jayawardene on eight.
Vandort, who resumed at his overnight score of 87, hit two consecutive sixes off left-arm spinner Abdur Razzak to move to 99 and then clipped two runs to mid-wicket in the same over to reach his third Test century.
Vandort put on 117 for the fourth wicket with Tillakaratne Dilshan when he was caught in the leg trap by Shahriar Nafees off spinner Moh