The Dons enjoyed the better of the chances with Gavin Rae guilty of heading wide in the best opening, but the Staggies defended resolutely to make it 36 league games without defeat.
A very healthy crowd of 14,010 were in attendance to witness the first ever competitive match between the sides.
Despite gifting Celtic a 1-0 win a week ago, Dons boss Craig Brown kept faith with goalkeeper Jamie Langfield, while defender Mark Reynolds, who has returned to Pittodrie on a season-long loan from Sheffield Wednesday, had to be content with a place on the bench.
Ross County boss Derek Adams, again without the suspended Stuart Kettlewell, made only one change to the side that opened their SPL campaign with a goalless draw against Motherwell with Sam Morrow replacing midfielder Russell Duncan.
A minute's applause was held before the start of the match in tribute to Aberdeen club stalwart Teddy Scott, who died in June at the age of 83.
The Dons, eager to avoid opening the new SPL campaign with back-to-back defeats, made an encouraging start with a clever flick by Scott Vernon giving Stephen Hughes a sight of goal but the midfielder fired into the side-netting.
The visitors had a penalty claim waved away by referee Euan Norris when Scott Boyd fell in the box under pressure from Dons skipper Russell Anderson, before Martin Scott passed up a glorious opportunity to break the deadlock when he sliced his shot wide with his left foot with only Langfield to beat.
However, it was the hosts who looked the more likely to open the scoring with County stopper Michael Fraser called upon to keep out a low Hughes effort before Rae thundered a shot over the crossbar from the edge of the area.
County captain Richard Brittain went close with a well-struck left-foot shot before the lively Jonny Hayes dragged an effort wide as the first half ended goalless.
The Dons should have made the breakthrough within six minutes of the restart when Gary Naysmith's back-post cross found Rae in acres of space, but the former Dundee midfielder somehow headed wide when it looked easier to score.
The home side were dealt a blow after 65 minutes when new signing Niall McGinn was stretchered off following an innocuous collision with Grant Munro and replaced by Ryan Fraser.
The Dons were dominating proceedings and threw on an extra attacker in Josh Magennis for Ryan Jack as they pushed for a winner in the closing stages.
However, it was the visitors who could have stolen the points with Brittain drawing a smart save from Langfield as the clock counted down.
Despite surviving a late scare when Vernon shot straight at Fraser from six yards out, County held on for a point as they proved they would be no pushovers this season with their second successive draw.