The tourists moved to 463-5 by lunch on the third day, gaining a massive lead of 231 over Bangladesh's first innings total of 232 and with five wickets in hand.
Jayawardene, Sri Lanka's leading run-getter in Tests with 11,116 runs, was approaching his 33rd century with a typically elegant innings that has so far contained six boundaries and two sixes.
Bangladesh failed to take a wicket in the morning session after Sri Lanka resumed on 375-5, as the bowlers struggled to contain runs on the even-paced pitch.
Sri Lanka's run spree on the second day revolved around a maiden Test century from opener Kaushal Silva (139) and useful contributions from Kumar Sangakkara (75) and Dimuth Karunaratne (53).
Also Read
Left-arm spinner Shakib Al Hasan has been Bangladesh's most successful bowler with three wickets, but his 37 overs so far have cost 129 runs.
Al-Amin Hossain has one for 100 in 30 overs, while fellow seamer Rubel Hossain has conceded 84 runs in 20 wicketless overs.