The Toffees earned a 1-1 draw at Arsenal on Sunday through a late equaliser from substitute Gerard Deulofeu, who is on loan from Barcelona.
It was a second positive result away to one of the leading Barclays Premier League clubs following on from the impressive 1-0 win at Manchester United in midweek.
Everton are now unbeaten in eight Premier League matches, and have climbed to fifth - just two points off second-placed Liverpool.
Osman, though, insists Roberto Martinez's well-drilled side will just continue to quietly go about their own business as they chase down a long-term ambition of qualification for the Champions League.
"We are a fit and strong team and have been going to these places having a lot of possession. When you do that you can dictate the pace of the game," said Osman, 32.
"We always came positive and wanting to win, but maybe there is a bit more belief now - however, we are not going to get ahead of ourselves.
"We have had a good week, but everyone knows things can change quickly.
"We will hopefully keep ticking along quietly and see where that takes us."
Martinez has done well since taking over from David Moyes, beating the Toffees' old manager on his first visit to Old Trafford.
While the Spaniard is determined to keep everyone at Goodison Park grounded, he believes recent results on the road have shown the current group have the opportunity to challenge the very best.
"At the moment, we are so focused on how we are trying to play that the next six months is going to be great potential for us as football club," Martinez said.
"We went to Manchester City earlier in the season and could not be ourselves, to play the way we wanted, and consequently we lost the game.
"Since then we went to Old Trafford and the Emirates in the space of a week and it was a completely different performance and that was an incredible sign.
"We have a blend which excites me and all the players are working hard trying to improve and develop.
"I am really excited about the potential of these group of players."
Martinez added: "Long-term Everton need to be in the Champions League, whether it is this season, I don't know, but we are going forward."
Midfielder James McCarthy collected his fifth booking of the season - which started playing for Wigan - on Sunday and so will now be suspended for the home game against Fulham.
Martinez said on evertontv: "It is one of those situations you have to cope with - it is the month of coping with the five bookings.
"We will have to make sure we use that time for him to recover.
"We have had a really intense week, but I thought James was superb once more."