The victory moved the Merseyside club out of the relegation zone and into 16th place with 21 points from 23 games.
The decisive moment of the match came in the 64th minute when Alex Iwobi sent Coleman to clear down the flank. Leeds goalkeeper Illan Meslier was expecting a cross, but Coleman arrowed his shot into the net from virtually the byline.
Despite creating numerous opportunities, Everton struggled to score, as has been the case for much of the season, with the league's lowest scorers lacking quality in the attacking third. The Toffees missed several chances to increase their lead but held on to claim all three points.
The win marks a successful start for new Everton coach Sean Dyche, who has guided the team to six points in three games since taking over. The victory also saw Everton move a point above the relegation zone, while Leeds slipped into the bottom three with 19 points.
Leeds failed to register a single shot on target during the game, extending their winless run to ten Premier League games, equalling their club record. The result may hasten their search for a new permanent manager, with Michael Skubala in temporary charge following the sacking of Jesse Marsch two weeks ago.
In the first half, Everton also had several chances to score, with Amadou Onana firing two shots wide from good positions. Leeds midfielder Weston McKennie also cleared off the line twice in quick succession, denying Everton the opportunity to take the lead earlier in the game.