The hosts recorded a 20-run victory over England to go alongside wins over the West Indies and New Zealand at the Super Eights stage.
Those results have resulted in many pundits tipping Sri Lanka to go all the way in front of their own supporters.
However, Sangakkara is keen for the squad to remain calm.
"It would be fantastic if we can win the tournament, [but] we've got to keep our feet on the ground and see what comes," he told Sky Sports News.
Sri Lanka have never previously won the competition.