Mendis (61) and Rathnayake (60) stitched together a 94-run stand for the third wicket after Sri Lanka were asked to bat, and skipper Shanaka provided the late flourish with a 19-ball 50 to take his side to 225-5 in 20 overs.
Shanaka's knock contained five huge sixes and two fours and eclipsed his Sri Lankan record of a 20-ball 50 against India in Pune three years ago, but he was caught in the deep off the bowling of Jiten Ramanandi after reaching the milestone.
"We're very pleased about the way we batted today," Shanaka said in his post-match interview.
"We used the conditions really well. Pavan and Kusal planned the innings really well, and helped myself and Kamindu Mendis to provide the finishing touches. I'm really happy with the batting and also with the bowling."
Playing without key spinner Wanindu Hasaranga, who was ruled out of the tournament with a hamstring injury sustained in their Group B victory over Ireland, Sri Lanka struck timely blows and never let Oman settle, restricting them to 120-9 in 20 overs.
Wasim Ali gave Oman hope with a breezy 27 alongside Mohammad Nadeem, who hit an unbeaten 53, but with wickets falling quickly at the other end there was no way back, and they fell to their second defeat after losing to Zimbabwe.
Dushmantha Chameera (2-19) and Maheesh Theekshana (2-11) led the way while Hasaranga's replacement Dushan Hemantha took 1-45 for a buoyant Sri Lanka, who will take on Australia at the same venue on Monday.
"It's very hard to replace someone like Wanindu because he's a champion bowler," Shanaka said.
"But Hemantha has been in the side for a while, playing a few games here and there in Wanindu's absence. So the confidence is there but coming into the game straightaway might be a bit harder. I'm sure he will come good.
"We will take confidence into our next game, with the batting, especially."