The hosts took just over an hour to claim the last five Bangladesh wickets with Jayawickrama claiming three of them for a second-innings bowling figure of 5-86.
Replying to Sri Lanka's first innings total of 493-7 declared, Bangladesh were all out for 251.
"There was a bit of pressure ahead of the debut but the captain and the senior players really helped me," Jayawickrama, 22, said after being adjudged player-of-the-match.
Home captain Dimuth Karunaratne heaped praise on Jayawickrama and young off-spinner Ramesh Mendis, who claimed four wickets in the second innings.
"Praveen was fantastic and Ramesh too," Karunaratne, who won the player-of-the-series award for his scores of 244, 118 and 66 in three innings, said.
"I think they need to play Test matches regularly and build confidence, and if they do that they will fill the shoes of Dilruwan (Perera) and Rangana (Herath) for sure."
"We lost this Test in the first innings when we were bowled out for 250. We should have batted better," he said.