Memphis Depay’s sixth-minute penalty gave Barcelona the lead after the Dutch forward was fouled by Nemanja Radoja.
Compatriot Luuk de Jong then raced on to Sergino Dest’s pass to score his first goal since joining on a season-long loan from Sevilla.
Substitute Ansu Fati hit the third on his return from a long-term injury.
Barcelona climb to fifth in La Liga and are five points behind leaders Real Madrid with a game in hand. Real were held by Villarreal on Saturday.
The pressure had been growing on manager Koeman after two wins from five La Liga matches this season had left them well adrift of the early pacesetters.
And although Koeman was absent from the dugout on Sunday, having been dismissed for dissent in Thursday’s goalless draw with Cadiz, his players had little trouble in putting an end to a three-game winless run in all competitions.
Levante failed to register a shot until the 81st minute, while Depay was thwarted by several good saves from Aitor Fernandez.
A return and goal for 18-year-old Fati – wearing Lionel Messi’s number 10 shirt on his first appearance since November 2020 after undergoing knee surgery – completed a fine success.