On numerous occasions this season, the Toffees have dominated matches but failed to capitalise and this appeared to be another instance.
However, the introduction of on-loan Real Madrid midfielder Royston Drenthe for the last 20 minutes of the Premier League clash at Goodison Park proved pivotal as it was his shot which Osman touched in.
Prior to that, the day seemed set to belong to burly striker Grant Holt, for whom the city of Liverpool has been a happy hunting ground.
The Canaries captain, who grabbed the equaliser against the Reds across Stanley Park in October, made it a Merseyside double with his 28th-minute goal.
Holt came off the bench to score at Anfield, which was the beginning of an eight-game spell which has now brought six goals.
Everton manager David Moyes can only dream of having a player in such form. His team's difficulties in front of goal have been a problem all season, as before kick-off only West Brom and Wigan had scored fewer.
The embodiment of that is midfielder Tim Cahill; usually a regular contributor up front the Australia international has now gone 363 days without a goal for the Toffees.
A tally of just 16 goals in 15 matches has undoubtedly contributed to a poor home record which has now brought only eight points from as many fixtures at Goodison.
That has placed additional significance on players to make chances count and it was highlighted in the first half.
Early on Cahill's touch-perfect lay-off from Leighton Baines' cross presented Marouane Fellaini with an opportunity from close range but he could not quite lift his shot over the body of diving goalkeeper John Ruddy, returning to his former club.
Osman's goalbound shot, deflected off Marc Tierney, was headed behind by Russell Martin and Osman's corner was only just flicked behind by Ruddy under pressure from Fellaini.
Despite having dominated it would have come as no surprise to regular Everton watchers when they went behind in the 28th minute from Norwich's first shot.
Phil Jagielka's header went straight up in the air and, when Steve Morison clipped the ball back in, Holt did well to hold off John Heitinga on the edge of the six-yard box before expertly turning the Dutchman to score off the inside of the far post.
Striker Louis Saha went close twice with a deflected shot which was clawed out by Ruddy almost under his own crossbar and a low angled drive just wide.
But hosts had a lucky escape late in the half when Phil Neville's slip on the edge of the penalty area put Wes Hoolahan in but no-one was able to take advantage of Andrew Surman's knockdown.
Everton started the second half on the back foot but created the first chance in the 54th minute when Ruddy saved Osman's left-footed shot.
But Holt was continuing to pose a threat at the other end and goalkeeper Tim Howard was relieved to see the striker's looping header drop wide of the post.
Cahill was replaced by Denis Stracqualursi on the hour and the Argentinian's first touch set up Saha for a shot which Ruddy parried to safety.
Then, with the goalkeeper off his line, the Frenchman had another effort blocked by Martin.
Everton's momentum was gradually growing but, considering their recent history, it was no reliable indicator that an equaliser was imminent.
Stracqualursi's introduction brought a more physical presence to the forward line but it was appearance of Drenthe, who has missed the last three matches with injury, which added the all-important injection of pace and unpredictability.
It was he who brought them a change in fortune as when he cut in from the right in the 81st minute to unleash a shot Osman got the slightest of touches to divert the ball inside the near post.
Drenthe's fierce, swerving shot then caused all kinds of problems for Ruddy, who was relieved to get something on the ball to turn it behind, before youngster Conor McAleny, on only his second senior appearance, snatched at his chance to win the match for Everton.