But Osbourne will take the positives into Wednesday's clash with St Johnstone as they seek to get their European challenge back on track.
Niall McGinn missed an early penalty and Aberdeen failed to capitalise on their pressure in the Clydesdale Bank Premier League clash at Pittodrie on Sunday.
Osbourne told RedTV: "In the changing room after the game you could see everybody's head was down but we have just got to get on with it.
"It happened earlier on in the season where we dominated games at home and only managed a point, but I feel we can hold our heads high because I thought we did quite well.
"It felt like we dominated possession and, considering the pitch, passed it around well.
"We had a whole heap of chances in the first half and we just have to be more clinical.
"If we had taken one chance I thought we could have scored three or four."
The Dons have taken two points from a possible nine in 2013 and St Johnstone can leapfrog them at McDiarmid Park on Wednesday night.
Osbourne said: "It's going to be a big game in the week but if we perform the way we did on Sunday I feel we will come away with a result and hopefully three points.
"I feel we need them because we want to push up the league and challenge for Europe.
"You can see a lot of the boys were very disappointed after the game but we just have to build on the performance and get a few results."
Osbourne was playing at centre-back against Hibs in the absence of the injured Russell Anderson.
"As long as I feel I can benefit the team, I will play wherever," he added.