The tournament minnows booked their spot in the round of 16 with a last-gasp 2-1 win over Austria in Paris.
iceland have advanced in unbeaten fashion, winning one and drawing two of their three Group F games, and will now face England in the next round.
Asked whether the win was a perfect riposte to Ronaldo's jibes, Bjarnason said: "It was great. It just shows what you can do when you have team spirit and not playing for yourself and it's unbelievable to be part of this great team.
"I'm really proud. It's an unbelievable moment to play with my good friends, I have played alongside them for many years, at Under-17s and Under-19s and now this, it's unreal.
"Looking up to the stands, 15,000 Icelandic people gathered, I never thought I could see the day and it's just a feeling of pride mainly.
"The belief in the team is tremendous. We've beaten big nations on the way here and got results against some of the biggest nations in the world like Portugal, Holland, Czech Republic and Turkey.
"The belief is there but we also know we have to hit 110 per cent on the day to get a result and hopefully it will be that day against England."
Like Iceland, England progressed as runners-up in their pool with a win and two draws to their name in Group B and Bjarnason recognises a strong outfit with a point to prove.
But Bjarnason is backing his team-mates to show their quality with the pressure off.
"They probably haven't played their best game yet in the tournament and they want to show and prove to everyone they have a good team," he added.
"If you see it player by player, they have one of the strongest sides in the cup and if they hit form on the day they're going to be hard to beat.
"With our work ethic, we will step up our game on the ball because going through was our goal and now we can play a bit more freely.
"We have to improve a lot on the ball and show what we did in the qualifiers, when we had possession a lot more."