The Toffees have spent the week soul-searching after their devastating loss to Wigan in the FA Cup quarter-finals but they looked back to their best in a fiery Barclays Premier League clash at Goodison Park.
Kevin Mirallas had a goal harshly disallowed before Osman scored from long range after 32 minutes and the Toffees held on despite the dismissal of Steven Pienaar and made sure through Jelavic in injury time.
All the urgency came from Everton as they demonstrated a hunger and desire they failed to show last week. Marouane Fellaini, himself subject of plenty of criticism this week, left no one in doubt as to his desire with a crunching challenge on James Milner. The downside for him and Everton was that it earned a 10th booking of the season and a two-match suspension.
City were slow to respond and thought they had been punished in the 13th minute when Kevin Mirallas lashed the ball into the net after Osman returned a clearance into the visitors' box. Much to City's relief, a flag had been raised for offside but replays suggested the decision was an extremely harsh one.
Everton were frustrated not to get a free-kick when Sylvain Distin was knocked over by Gareth Barry in a challenge and the City midfielder was caught soon after by Pienaar, who earned his first yellow.
Aleksandar Kolarov was the next to be booked after up-ending Osman, whose next significant contribution was to fire the Toffees ahead. The midfielder, whose form earned another England call-up this week, took a square ball from Seamus Coleman 25 yards out and struck a first-time shot which swerved viciously away from Joe Hart.
Everton went close again when Osman had a volley deflected over from a corner and the feisty atmosphere of the first half returned when Pienaar caught Javi Garcia and was shown his second yellow card.
City continued to press and had an opportunity when a Carlos Tevez shot struck Fellaini's arm in the box, but a free-kick outside the area was awarded and Kolarov struck the wall.
Everton rode out the storm and broke away in injury time through Fellaini, who rolled wide for Jelavic to beat Hart with a deflected shot.
Source: PA
Source: PA