Everton have exceeded expectations this season and will end the year in fourth place, just five points adrift of Premier League leaders Arsenal.
Liverpool and Manchester United would appear more likely contenders for a Champions League place, but Osman urged his team-mates to set their sights high in 2014.
"I think you have to aim for it - if you are not, you are not really setting yourself any targets," said Osman. "We are certainly aiming high.
"We are still up in the fight and we are aiming to try to get as many points as we can. We are ticking along nicely."
Osman made his 303rd Premier League appearance for the Toffees against Saints, a new club record as he overtook David Unsworth.
It had been a swift change of fortunes for Osman, whose blunder led to Sunderland's winning goal in the 1-0 Boxing Day defeat.
The midfielder, who was named as skipper, said: "It feels great, and to do it on a day when we got an important victory makes it even better.
"It is just one of those things that Jags (Phil Jagielka) missed out, and I was next in line to get the armband, but it was an extra honour to wear it this time.
"As everyone knows, the last game was disappointing, especially for myself.
"So it shows how quickly things can change in football - against Southampton, I was captain of a winning team on a record-breaking day."