The visitors looked set to complete a smash-and-grab raid after Regan Charles-Cook's first-half strike, only for American Ramirez to force home an 88th-minute equaliser.
The Dons announced the signing of Scotland centre-back David Bates from Hamburg before the match and handed debuts to new loan signings Austin Samuels and Matty Longstaff, while the visitors were unchanged from their defeat to Rangers a week ago.
Aberdeen looked threatening from set-pieces, with Calvin Ramsay's deliveries causing panic in the visitors defence as early as the third minute when Declan Gallagher had a shot blocked.
The Dons have posed a threat going forward but have been easily opened up at the back this season and almost shot themselves in the foot again when Scott Brown's back pass was intercepted by Jordan White.
He rolled in Ross Callachan, but goalkeeper Joe Lewis was off his line quickly to smother.
Brown had his heart in his mouth again shortly afterwards when he nudged a Connor Randall cutback narrowly past his own goal for a corner.
The former Celtic skipper had a big penalty claim at the other end when his shot from another Ramsay corner seemed to strike a hand in the box, but referee Nick Walsh waved away the strong appeals.
It was County who took the lead to silence the 14,434 crowd in Pittodrie.
Again defending was slack as Blair Spittal's left-wing cross saw Harrison Paton's shot blocked but Charles-Cook followed up to tuck home the loose ball.
They were unlucky not to level on the hour when Samuels' cross from the left was begging for a touch but was eventually gratefully dived upon by Ross Laidlaw.
That would be Samuels' last involvement as he was withdrawn moments later, along with the hard-working Longstaff, for Marley Watkins and Jay Emmanuel-Thomas.
County came close to doubling their lead after 69 minutes as Callachan got in behind the defence and rounded Lewis only for Ramsay to clear off the line, and at the other end Ramirez missed his kick with the goal at his mercy.
Alex Iacovitti was well-placed to turn behind a Lewis Ferguson header from another Ramsay cross, and the home side piled on the pressure as the clock went against them, but the leveller eventually came with just two minutes left.
Laidlaw failed to hold onto a Jonny Hayes shot and as the ball squirmed towards the goal-line, Ramirez slid in to force the ball home and deservedly level the scores.