The tourists created a bit of history, when they completed a 121-run victory over Chris Gayle's side in the opening Test on Wednesday.
"We had a challenge when we came and that was to win a Test match in the West Indies and we have achieved that," Jayawardene told reporters.
"We played hard and a lot of guys contributed. It was an amazing effort from Chaminda Vaas, and our 'quicks' took 13 wickets in the Test, so a lot of credit to them. Muttiah Muralitharan bowled his heart out, and overall it was a very good team effort."
Jayawardene felt West Indies played into his side's hands, when they chose to chase an unlikely 437 to win.
"We wanted them to play a few shots on this pitch because that's how we could have created opportunities," he said.
"That's why we gave them a target they could have achieved, rather than giving them 500 when they would have been in a very defensive frame of mind. We gave them a target thinking this is a target they could chase especially since they have shot-makers."
Jayawardene indicated when Dwayne Bravo and Ramnaresh Sarwan were at the height of a record second wicket stand of 134, he had some reservations about if his side could complete the job, but they stuck to the task.
"The last hour of the fourth day, their attitude was totally different in the way they approached it, and we knew that we were up against a good challenge," he said.
"Bravo and Sarwan both batted really well and we had to be very patient with those two guys, make sure that we don't give them too many easy runs, wait for our opportunity, and once we got the breakthrough, pick up a few more wickets, and put pressure on the other batsmen.
"If Bravo and Sarwan had batted until about tea time, it would have been a different story. We would have struggled, but we managed to get Bravo before lunch, and then Sarwan before tea which were two crucial wickets, and in between we got Shivnarine Chanderpaul which was an important wicket for us."
Jayawardene explained that though Gayle had dropped himself down the order, his side always felt they just need to put pressure on the other batsmen and the results would come.
West Indies captain Chris Gayle was disappointed that his side could not hold on to draw the Test and keep the two-Test series level at 0-0.
"It was tough coming out on the losing side again, but the guys really fought well to try and save the match," he said.
"We have to give them credit for this, since the conditions were tailor-made for the Sri Lankans, but there are a few positive that we can take away from this game."
Gayle's decision to bat down the order in the West Indies' second innings and allow Bravo to open the batting created plenty of discussion among the home team's fans, but he asserted this was not a long-term measure.
"When Bravo was bowling, I walked up to him and jokingly asked if he wanted to open the batting," he said.
"He replied yes because he felt it would have been a good move for him too to avoid having to face the spin bowlers right away, and get into his groove against the fast bowlers first.
Gayle admitted that Vaas had the upper-hand on him, but this was just another challenge he would have to confront in his international career.
The second Test at Queen's Park Oval in the Trinidad capital of Port of Spain begins on April 3.
Sri Lanka, eyeing a Test series victory for the first time in the Caribbean, have lost two and drawn two of their previous four Tests in this part of the World.