Jagielka was injured by a challenge from Adam Le Fondre and manager David Moyes said he was going straight to hospital.
"I'm disappointed because Jags is such a good player," Moyes was quoted as saying by the BBC. "It's a bad one. He'll have surgery tonight to stitch it up. I've genuinely not seen the challenge again. At the time I thought it was poor.
"Maybe if I see it again I will think differently."
Jagielka, 30, has made 16 appearances for England and his injury could jeopardise his involvement in this month's World Cup qualifiers away to San Marino and Montenegro.
Goals from Marouane Fellaini, Steven Pienaar and Kevin Mirallas earned the points for Everton who maintained their push for a top-four finish.
They are now six points behind Tottenham Hotspur who are fourth and face fifth-placed Arsenal on Sunday.