Following Liverpool's 4-1 defeat at Arsenal earlier on Saturday, Ronald Koeman's Southampton had the chance to move up to fifth with a win at Goodison Park.
But although they made a promising start, the visitors fell behind against the run of play when Everton skipper Jagielka scored a scrappy opener - shortly after Tim Howard had pulled off a wonderful save to deny Graziano Pelle a first Premier League goal of 2015.
James Ward-Prowse almost restored parity with a curling free-kick but, despite their domination, Southampton could not force an equaliser before the break.
The second half continued in a similar fashion, with substitute Dusan Tadic Southampton's main threat, but Everton held on for their third top-flight win in a row as Koeman's side missed the chance to overtake Liverpool in the race for UEFA Europa League qualification.
Already shorn of top scorer Romelu Lukaku due to a knock, Everton were dealt another blow prior to kick-off when it was revealed that Darron Gibson will be out for the foreseeable future with a fractured metatarsal.
However, Martinez was able to welcome back Gareth Barry from suspension, while Ross Barkley also came into the starting line-up.
But it was Southampton who started on top, Howard doing superbly to claw an effort from Pelle - whose last league goal came in the reverse fixture in December - off the line.
That chance seemed to spark the hosts into life, James McCarthy going close before Jagielka nudged them ahead after 16 minutes.
Southampton's usually rock-solid defence was a shambles, as they squandered several opportunities to clear from a corner, allowing Barry to square for Jagielka, who duly prodded home from close range.
The visitors came back strongly, with Ward-Prowse twice going close - first firing a free-kick just wide of the post before then forcing Howard into another fine save with a rasping long-range drive.
Koeman turned to Tadic and Sadio Mane for inspiration with half an hour remaining, and Tadic was involved immediately as he whipped in a superb corner towards the head of Shane Long, only for the striker to mistime his jump and direct a header wide.
In response, Martinez introduced Muhamed Besic and Steven Naismith, with the latter having two penalty appeals waved away by referee Lee Mason within a minute of coming on.
Everton showed little attacking intent as the game drew to a close, but hung on to secure the spoils and move up to 11th, while Southampton's hopes of European qualification hang in the balance.
