Buttler made an unbeaten 73 and captain Morgan hit 47 not out as the hosts eased to their target of 141 with more than two overs to spare.
England, who won the test series between the teams 2-0 and took a rain-disrupted one-day international contest 3-0, continued their complete dominance of the Sri Lankans with a ruthless performance.
Sri Lanka never looked like posting an imposing total after winning the toss, opener Dhanushka Gunathilaka top-scoring with 26 as England took wickets at regular intervals.
Left-arm spinner Liam Dawson completed excellent figures of three for 27 on his debut and the hosts looked sharp in the field, good work by Jason Roy and James Vince leading to run-outs.
England lost Roy in the first over and Vince fell for 16 but Buttler, promoted to open the innings, struck four sixes and three fours to keep his team in control and Morgan clinched the win with a big six over mid-wicket off Ramith Rambukwella.
"I'll bat wherever the team wants me to bat. It's something I really want to develop, the longer I can bat, the more runs I can get," Buttler said.
Morgan was delighted with his team's performance in the field.
"Tonight, our fielding peaked, we were hitting the stumps for fun. We've spoken about fielding, it's the one thing in cricket you can control," he said.
"We have athletes in the field and we must use those as much as we can."
Sri Lanka captain Angelo Mathews reflected on a disappointing tour.
"We showed fight in bits and pieces. We were not consistent enough," he said.
"That is something we need to work on. There are positives. Unfortunately we didn't have the bowling to put pressure on."
