The 3-0 victory over Newcastle, their first home league win of 2015, was vital in restoring a six-point cushion between themselves and the bottom three.
It relieved some of the pressure which was reaching crisis level among some fans, and after beating Dynamo Kiev last Thursday back-to-back wins for the first time since November have helped lift the gathering gloom.
"We really needed the win
We had a good result in Europe in the week but it was really vital we turned up and got three points in the league, and we did that really well," said Osman, on his first league start since December 3 after returning from injury.
"We are not trying to look down, we are trying to look up but out the corner of your eye you can't help but see how close you are (to the bottom three) at times.
"With Burnley maybe surprisingly beating Man City on Saturday it was vital we won.
"We've not been the best in the league of late, so it was vital we got ourselves back on track.
"On occasions things don't always go the way you want to on the pitch but for us it is not for a lack of trying, we tried the same as we have in the last few weeks.
"We seemed to get things right as a team and we pressed together as a team and things came off for us.
"Confidence is massive: you can be the best team in the world on paper but if you haven't got confidence you are not going to get a result.
"I think maybe we lost a bit of confidence over the Christmas period, but over the last few weeks we started to get it back and that showed out there in the last two matches."
Despite it only being two results there finally seems to be a feeling of momentum gathering at Goodison Park, but that needs to continue in Ukraine this week for their last-16 second leg tie, in which they lead 2-1.
"We have been good in Europe this season, especially away from home, so we are going there with confidence but these ties in countries like Ukraine are difficult," added Osman.
"We are ticking along nicely and starting to get our confidence back with two wins out of two and hopefully we will make it three on Thursday."
Manager Roberto Martinez made five changes for the Newcastle match and he will rotate players again for the trip to eastern Europe.
The injury list is reducing after left-back Leighton Baines returned against the Magpies having missed the previous five games, with forward Kevin Mirallas available to face Kiev, and Osman believes that should help them finish the season strongly.
"There is a lot of competition for places but we are in it as a team: there is no sulking if you are not in the team at the minute, it is all about the 11 out there getting a result," he said.
Newcastle, by contrast, are seeing their options diminish by the week and captain Fabricio Coloccini's suspension for a red card at Goodison adds another name to the list of those unavailable.
"Just when you think you are getting somewhere on the squad front in getting Jack Colback back from suspension and Remy Cabella and Robbie Elliott back into the group, all of a sudden you lose your captain for three games and a defender which, as we all know, we are not blessed with many," said caretaker boss John Carver.
"I can't bring out a magic wand and bring in anyone from the outside but I am losing people not through silly training ground injuries or silly red cards.
"I have a huge injury list and I am working with what I have got
I know what situation I am in and I can't do anything about it."
Source : PA
Source: PA