The home side had trailed 16-3 during the first half as Johan Goosen kicked three penalties and converted Francois van der Merwe’s try.
But they roared back through three Dan Biggar penalties and a tremendous late try from Josh Matavesi.
Goosen had a chance to give Racing all four points with a late penalty attempt, but his long-range effort fell wide as an engrossing clash finished all square.
The Ospreys were missing a host of forwards through injury and Racing ruthlessly targeted that area during the first half.
A powerful rolling maul and a destructive scrum shove allowed Goosen to slot two early penalties for the visitors, although Biggar responded with one of his own.
But Racing took control when they registered the first try. They worked Van der Merwe into space and the giant lock barrelled his way over the line, with Goosen converting.
Another penalty from the South African made it 16-3, although Biggar did chip away at the gap with his second penalty.
However, the Welsh side survived a mighty scare right on half-time when Racing centre Alexandre Dumoulin dropped an inside pass from Marc Andreu which would have sent him under the posts.
The Ospreys made the most of the lifeline, with scrum-half Rhys Webb and Biggar injecting real tempo into their play for the first time in the game.
Biggar missed one easy penalty chance, but slotted two more to get the Welsh side to within four points. Racing returned to their forward strengths and worked a drop goal chance which Goosen took to make it 19-12.
But the hosts responded brilliantly. Justin Tipuric broke through and Webb seized on his offload to send Matavesi in, with Biggar’s conversion tying the scores. Goosen had a late chance to snatch victory with a long-range penalty, but he was off target in a tense finale.
