Momentum swung firmly from the hosts to Sri Lanka on Monday at Headingley on day four as the tourists added 243 runs to their overnight total before reducing England to 57-5 in pursuit of 350 to win.
Mathews was inspirational with the bat as he reached a career best 160 thanks to 26 boundaries before being dismissed by James Anderson (3-91) as England toiled.
Dhammika Prasad then ripped through the England second innings with an impressive six-over spell that saw him claim four wickets for just 15 runs.
Sri Lanka now stand on the brink of a first series win in England since 1998.
"It will be a successful tour if we can pull it off, (but) it's a job only half done, we'll stick to the basics," skipper Mathews told Sky Sports.
"On a fifth day there's a bit of turn and help for the spinners so obviously (Rangana) Herath is going to come to the party.
"After the first couple of days no-one would have thought we'd be in this position
"It's a bit of an up and down wicket but we tried to stay positive and score runs."