The Merseysiders had to grind out victory against a well-drilled but luckless Wigan side in heavy rain at Goodison Park on Boxing Day.
A deflected strike from Osman and a Phil Jagielka header gave the Toffees what proved a decisive lead, but they needed to hang on for a 2-1 win after Arouna Kone poked in a scrappy late reply.
The win over the struggling Latics kept David Moyes' men joint fourth in the Barclays Premier League and in good heart for the different challenge presented by third-placed Chelsea this weekend.
Osman said: "It is always a little more difficult at Christmas time with people's moods and the way the weather was, so it was a terrific three points to dig it out against a good Wigan side.
"Chelsea are coming here on Sunday full of form now but we are confident and hopefully we will make a good game of it.
"We are certainly a developing team and have developed a lot over the last year or so.
"We are playing some good football and we tend to show that more when we play against the teams at the top end of the table."
Victor Anichebe blazed a good chance wide for Everton after just 13 seconds but Wigan went on to control large spells of the first half with little reward.
Everton upped the tempo after the break and hit the bar through Thomas Hitzlsperger before Osman broke the deadlock in the 52nd minute with a shot which went in off Gary Caldwell's arm.
Osman, 31, said: "The rules say when you hit a shot and it's on target it's your goal.
"Obviously it did take a deflection and the keeper may well have saved it without the deflection but it's gone in and I'm happy we opened the scoring."
Osman was then involved in a controversial incident as Shaun Maloney went down under his challenge in the area, but referee Lee Mason gave no penalty - much to the visitors' anger.
"It is ultimately what the ref sees and the ref decided not to give it," said Osman, when asked about that incident.
Everton doubled their lead when Jagielka crashed in a header off the bar from a cross by Phil Neville, who was playing his 500th Premier League game.
Despite that second blow, Wigan refused to capitulate and belied their lowly position by continuing to play enterprising football.
They made life uncomfortable for the hosts after Kone pounced on defensive uncertainty eight minutes from time, but Everton held on to extend their unbeaten run to seven games.
Osman, who won his first England cap last month, said: "If you've watched Wigan at all you'd know that they do that everywhere they go.
"They go and play their football and mostly outpass teams. We knew they were going to do that again.
"It is all about getting three points at the end of the day. We knew there would be times when we'd have to sacrifice possession but if we took our opportunities we'd be fine, and we did that."
Despite their positive play, the defeat represented a sixth successive game without a win for Wigan and left them 18th in the table.
After near misses in the past two seasons, Roberto Martinez's men look to be heading into another relegation fight.
Martinez has no intention of abandoning a footballing philosophy which has ultimately been vindicated in each of those successful battles against the drop, but he wants more end product.
The Spaniard said: "We need to improve on what we do.
"We play well but it is important we get better in both boxes.
"We can't be conceding goals in every game, even if - like the first one - you can't do much about it.
"In football you need to be stubborn in what you do but we must make sure we change the dynamics to get that first good result that would allow us to go into a good run.
"We are playing well enough and we are playing full of character and determination.
"It would have been all too easy once we went 2-0 down not to show spirit but it couldn't be further from that.
"That is a real positive we are going to take into the next game."