Ashley Williams did hack a Victor Anichebe effort off the line as Everton pushed for a first-half breakthrough, and the in-form Michu saw an attempted lob tipped onto the crossbar by Tim Howard at the other end, but there were few clear-cut opportunities created in 90 minutes.
Everton were boosted by the presence of Leighton Baines, who overcame an ankle problem, while Swansea freshened up the side which claimed a Capital One Cup victory over Chelsea in midweek with Michel Vorm, Nathan Dyer, Dwight Tiendalli and Kemy Agustien.
Marouane Fellaini had the first chance of the game for Everton inside the opening minute, but his 20-yard shot was easily claimed by Vorm.
Vorm then needed to be alert after a poor back-pass from Ben Davies allowed Nikica Jelavic in, but the goalkeeper blocked and then cleared as Anichebe threatened.
Swansea had their first serious opportunity when Pablo Hernandez launched a counter-attack, but his ball inside for Nathan Dyer was too strong and Howard gathered.
Hernandez, one of four Swansea Spaniards hoping to impress in front of national coach Vicente del Bosque, then attempted a spectacular shot from outside the area but screwed his effort well wide.
Anichebe went close when he reacted quickly after Fellaini headed on a Phil Jagielka ball, but Vorm palmed away his header and Williams hacked off the line.
Michu was inches away from giving Swansea the lead as he raced onto a fine ball from Dyer and delicately attempted to chip Howard from close range. Howard backtracked and managed to palm the Spaniard's fine effort onto the bar.
Everton made a positive start to the second period as Phil Neville took up an attack from Seamus Coleman and forced Vorm to save a low shot.
The Toffees kept up the pressure with Fellaini winning a loose ball and Anichebe lifting the ball across the box, but Jelavic failed to make contact with an attempted overhead kick.
The game became rather scrappy as it wore on but Baines still offered quality and he presented Jagielka with a fine chance from a corner 16 minutes from time, but the centre-back headed over.
Everton sent on Steven Naismith in a late attacking change but were unable to break through with Fellaini heading at Vorm but being penalised for a push and Jelavic shooting over.