The pair added an unbroken 212 for the second-wicket to help their side win by nine wickets in the 48th over. Thirimanne finished unbeaten on 139 off 143 balls, with Sangakkara on 117 off 86 - his second century in as many matches.
England captain Eoin Morgan admitted afterwards that his bowlers simply hadn't got things right, saying, "Today we bowled a bad ball every over which you get punished for. When we are bowling well, it is a good attack. Today they didn't."
Only Moeen Ali took a wicket, to end a 100-run opening stand by having Tillakaratne Dilshan caught by Morgan for 44. The seam bowlers failed to take a wicket in 35.2 overs between them. All four conceded at least six runs per over.
The match had actually started well for England as they won the toss and got off to a brisk start courtesy of Ian Bell. Sri Lanka missed a couple of difficult chances early and Bell looked like he was going to make full use of them.
However, the change bowlers - led by Angelo Mathews and Dilshan - stifled the charge with Moeen dismissed for 15 with the score on 62 and the out-of-form Gary Ballance adding to his problems by being caught and bowled for just six. When Bell was then bowled by the returning Suranga Lakmal for 49, it left the scoreboard reading 101 for three.
Eoin Morgan and Joe Root began a recovery of sorts by adding 60 for the fourth-wicket, but a scratchy innings from the skipper was ended by Thisara Perera at 27 to leave the lower-order with plenty to do. Fortunately for England, after a period of relative calm, they prospered. James Taylor was typically busy in making 25 and accompanied Root through to his fourth ODI century before being caught off Lasith Malinga.
Jos Buttler then added his contribution by hitting seven boundaries from 19 balls to finish on 39 as England passed 300 - a score that was probably only about par on a flat Wellington pitch. The end of the innings threatened to descend into farce from a Sri Lankan perspective as they used four bowlers in the final two overs. Rangana Herath was forced off by injury in the 49th, while Lakmal was removed four balls into the 50th after bowling two beamers.
Thirmanne and Sangakkara would soon ensure that England wouldn't even have a chance to bowl the last two overs of their innings as their whole tournament continued on a trajectory that many could quite easily describe as farcical.