Jayasuriya scored 74 on the fourth day of the Test on Saturday as Sri Lanka set the tourists a tricky 325-run target on a surface that is offering extravagant turn and becoming increasingly variable in bounce.
"We have a very good chance because it is not easy batting out there now," Jayasuriya said on Saturday. "The ball is starting to turn a lot and it is not easy to get runs.
"It will be very difficult for their batsmen tomorrow.
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"We took a longer time to bowl them out in the first innings but Upul (Chandana) and Murali didn't bowl so well then.
"If one of them bowls really well tomorrow, we will win."
Jacques Rudolph, meanwhile, who scored 102 as South Africa were bowled out for 376, believes his team can still win.
"It is going to be interesting tomorrow," said Rudolph. "If we can have a good first session tomorrow that will probably dictate whether it is going to be a win for us or not."
Rudolph's hundred was the fourth of his career, and the latest sign that he could fill the void left by the retirement of Gary Kirsten.
"I really had to occupy the crease and spend some time out there. It is a special feeling. It is always nice to get a Test hundred, especially here in Galle against the likes of Muralitharan.
"He (Kirsten) left a legacy that is going to be hard to fill, but I want to fill that same kind of role. When the going gets tough, I want to be in there."
