Jagielka volleyed home from Leighton Baines' 36th-minute corner to complete a repair job after Leon Best had headed the Magpies in front 13 minutes earlier.
Osman started the fightback when he levelled with 31 minutes gone after exchanging passes with the excellent Mikel Arteta on an afternoon when the home side struggled to find any real fluency.
Best thought he had snatched a point 12 minutes from time when he headed past Tim Howard for a second time, but referee Howard Webb had spotted a push on Jagielka as the ball came across and disallowed it.
Newcastle battled all the way to the whistle in a desperate attempt to rescue the game in front of a crowd of 50,128, but Everton were good value for their win.
Alan Pardew's side went into the game knowing victory would all but assure them of their place in next season's Premier League.
They were buoyed too by the knowledge that the visitors would be without key men Marouane Fellaini and Tim Cahill, although their optimism was tempered by the absence of both Joey Barton and Stephen Ireland through injury, which presented Pardew with a problem on the right side of midfield.
His initial solution was to promote full-back Danny Simpson to the role and, after a scrappy start, it was the Magpies who took the lead from their first meaningful attack with 23 minutes gone.
Fabricio Coloccini's quickly-taken free-kick to Jonas Gutierrez caught the visitors napping and when the Argentinian's low cross was turned on to Kevin Nolan by Mike Williamson, he powered a shot towards goal from a tight angle.
Howard could only parry the ball across goal to present Best with the simplest of tasks to head home his sixth of the season.
The goal seemed to settle Newcastle down and they pinned the Merseysiders back as they sought to increase their advantage.
However, their dominance was fleeting and the lead not only evaporated, but was overhauled in the space of five minutes.
Arteta, playing on the left side of the Everton midfield, had earlier shown signs of causing problems, and he did the damage with 31 minutes gone.
He was well-placed to receive Osman's pass after his midfield colleague had broken from halfway, and laid the return ball on a plate for him to side-foot home with the help of a deflection off Jose Enrique.
But there was worse to come with 36 minutes gone when the black and white shirts stood and watched as Baines' free-kick sailed across the penalty area for Jagielka to shoot home off the underside of the crossbar.
Newcastle were unable to contain the visitors in the middle of the park and found Jermaine Beckford, once a target for former boss Chris Hughton, repeatedly running at defenders with Arteta prompting.
Pardew was forced into a reshuffle when Enrique limped off injured and with Simpson taking over at left-back, Peter Lovenkrands dropping into midfield and masked substitute Shola Ameobi joining Best in attack, they had a better shape.
They might have gone in at the break level had Williamson managed to turn Gutierrez's driven shot inside rather than outside the post in injury time, but even then, Pardew would have had work to do anyway after a disjointed opening 45 minutes.
Ameobi started the second half in positive mood and twice within a minute forced saves from Howard, the first of them less than convincing.
But with John Heitinga having replaced skipper Phil Neville, Everton could have increased their lead with Steve Harper saving from Beckford and Louis Saha heading wastefully over after Arteta had once again sliced his way into the penalty area.
The game was becoming increasingly open to leave both defences under pressure, and Harper had to make a smart save to keep out Beckford's 61st-minute volley.
Arteta might have sealed the win with 16 minutes remaining, but stabbed wide from close-range, and Best had a 78th-minute header chalked off for a push on Jagielka.
The Magpies refused to throw in the towel, but simply did not have the craft or guile to get themselves back on terms, and substitute Seamus Coleman might have increased the visitors' lead at the death but for a good block by Harper.