Swansea coach Paul Clement was forced into an early substitution as Wilfried Bony suffered injury in the fifth minute and was replaced by Tammy Abraham.
The Swans pressed the hosts high up the pitch after as they looked to unsettle Sam Allardyce's charges who were building from the back.
In the 29th minute, Aaron Lennon pulled off a beautiful turn inside the box and fired a well struck shot that sailed inches wide of the upright.
Leroy Fer opened the scoring for the visitors in the 35th minute with a simple finish from close range after getting on the end of Tom Carroll's corner.
The Swans had the momentum after 40 minutes and were threatening to score the second goal before halftime but the home side survived the spell of attack.
The Toffees were awarded a penalty in the second minute of first-half stoppage time. Wayne Rooney's spot-kick was saved by Lukasz Fabianski but Dominic Calvert-Lewin reacted the quickest to score from the rebound, 1-1.
Allardyce made his first alteration in the match 10 minutes after the break, as he took off Morgan Schneiderlin for Tom Davies.
Gylfi Sigurdsson put the Toffees ahead four minutes after the hour mark after cutting onto his favoured right foot and curled a fine strike into the top corner, 2-1.
With 15 minutes remaining, Rooney got another chance to strike from the penalty spot and this time the forward managed to convert to extend the hosts' lead, 3-1.
The Toffees were not in a rush to go forward in the last five minutes of the match as they enjoyed a two-goal cushion while the Swans looked out of steam.
The win sees the Toffees move up to ninth in the Premier League standings, while Swansea remain rooted to the bottom.