Resuming on 57 for five with an unlikely 350 required for victory, England made a positive start to the last day of the second and final test but lost Joe Root (31) and Matt Prior (10) as they reached the interval on 173 for seven.
Prasad took his fifth wicket of the innings when he got rid of Prior, leaving Moeen Ali, unbeaten on 56, and Chris Jordan, nine, to try to salvage a draw for the hosts.
Earlier, perhaps frustrated by the lack of a breakthrough, Sri Lanka captain Angelo Mathews exchanged words with Root and umpire Billy Bowden was forced to intervene.
As the sledging continued Shaminda Eranga subjected Root to a feisty spell, troubling the 23-year-old with some fiery bouncers.
Root's defiance was finally ended when he edged Nuwan Pradeep to Lahiru Thirimanne in the gully, bringing an end to a 67-run partnership with Moeen for the sixth wicket.
The first test at Lord's ended in a draw.
