Osman scored a deflected opener early in the second half before Phil Neville marked his 500th Premier League appearance by setting up Jagielka for the second.
Arouna Kone gave Wigan a late lifeline but it was not enough, the visitors having earlier been denied a penalty when Shaun Maloney went down in a challenge from Osman.
Everton almost made the perfect start as Victor Anichebe shot wide after just 13 seconds, while James McCarthy had a powerful effort blocked for Wigan.
Leighton Baines set up Steven Pienaar but Ali Al Habsi saved well, and Anichebe then headed Thomas Hitzlsperger's cross wide.
Everton started the second half with renewed vigour and Thomas Hitzlsperger struck a thunderous long-range effort against the bar after nutmegging McCarthy.
Osman got lucky after 52 minutes, his shot from just outside the area striking Gary Caldwell's arm and wrong-footing the unfortunate keeper.
There was controversy as Wigan tried to respond, as Maloney went to ground in the area in a challenge from Osman. The Latics were incensed not to be awarded a penalty, with Jean Beausejour booked, but replays suggested Maloney may have sought contact.
Everton doubled their lead when Neville received a short corner and sent in an inch-perfect cross for Jagielka, who powered in a header off the bar.
Wigan fought on and gave themselves late hope when Kone bundled the ball in at close range after Sylvain Distin and Baines failed to clear from a Maloney shot, but they could not find an equaliser.