He claimed match figures of 11 for 178, the best by a Sri Lankan spinner on debut.
The spin twins pick five wickets between them as hosts eye series win on final day.
A dream debut for Praveen Jayawickrama helped Sri Lanka restrict Bangladesh to 251/10, extending their first-innings lead to 259 runs at stumps.
Bangladesh bowlers fashioned a middle-order collapse for Sri Lanka, before an unbroken 87-run seventh-wicket stand steadied the hosts.
Centuries from openers steered Sri Lanka towards a potentially mammoth first-innings total on day one of the final test in Pallekele.
Asolid opening stand between Dimuth Karunaratne (85*) and Lahiru Thirimanne (58) headlined Sri Lankans strong response.