Sangakkara hit an unbeaten 117 as the 1996 champions thrashed England by nine wickets in Wellington, following up on his 105 not out against Bangladesh.
It was Sangakkara's 24th century in 402 one-day internationals and Thirimanne said the vastly experienced 37-year-old was a steadying influence on the team.
Asked where Sangakkara, who is playing in his final World Cup, ranked in the pantheon of greats, Thirimanne said he was right at the top.
"Not only is he rotating the strike, but hitting the boundaries as well. So it is very helpful for the team."
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Thirimanne himself made an unbeaten 139 against England and said he was thrilled to carry his bat after only being promoted to open in the lead-up to the tournament.
He also has two half centuries at the tournament, while Sri Lanka's potent batting line up includes another two World Cup centurions in Tillakaratne Dilshan and Mahela Jayawardene.
But Thirimanne said Sri Lanka, finalists at the last tow World Cups, could not rely on batting alone and had to improve other aspects of their game.
"You have to field, you have to bowl really well. So we have to improve on that side.