With England manager Gareth Southgate watching from the stands, Everton keeper Jordan Pickford was in fine form, blocking a shot from Patrick Bamford with his legs and diving at full stretch to deny Raphinha.
Everton's James Rodriguez and Richarlison had the ball in the net in the first half but both efforts were ruled out for offside, and Leeds keeper Illan Meslier made superb saves from Mason Holgate and Richarlison to keep his clean sheet intact.
Jack Harrison went closest for Leeds, smacking a header off a post as the first half ended goalless, and Meslier was almost caught out early in the second period when he passed the ball to Rodriguez but recovered to catch the Colombian's tame lob.
There was no let-up in the action and Bamford had a goal ruled out for offside in the 65th minute before Raphinha took centre stage.
The Brazilian, signed from Rennes in October, appeared to be looking for a pass but he checked back, driving the ball between the legs of defender Ben Godfrey and in at the foot of the far post in the 79th minute.
The two teams managed 38 goal attempts in the match but that proved the only one to count and the victory lifted Leeds to 11th place in the table on 14 points. Everton are in sixth place on 16 points.
"It was a very entertaining game," Leeds manager Mario Bielsa told Sky Sports.
"The result was a fair one and the performance of our team was a satisfactory one."