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Barkley's counterpart, Ireland fly-half Ronan O'Gara, knocked over an early penalty as Munster dominated early play.
It wasn't until the quarter-hour mark, however, that the Irish province scored their first try, flanker Sean Dougall bouncing off a couple of weak tackles to crash over from short distance, O'Gara adding the extras.
Barkley, making his first appearance for Racing since joining from Bath earlier this month, pulled back three points with a long-range penalty.
Against the run of play, a fumbled O'Gara pass by Munster skipper Doug Howlett, the former All Black, saw Racing counterpart Jacque Cronje hack through.
Home scrum-half Maxime Machenaud was on hand to race through and touch down for a try Barkley converted to draw the scores level.
O'Gara limped off just after the half hour, and half-back partner Connor Murray was lucky to escape a sin-binning after taking out Karim Ghezal off the ball as he chased through a charged-down kick with the line begging.
Barkley was successful with two more penalties either side of the break, but went wide with his fourth effort.
Munster full-back Ian Keatley then saw his attempt miss, but wing Simon Zebo provided some much-needed impetus after a quick play of the ball at a lineout on the halfway line, outpacing Mirco Bergamasco and cutting inside Julien Jane for a memorable try.
Keatley knocked over the conversion to hand Munster a 17-16 lead, but shoddy defensive coverage of an innocuous kick by Murray deep inside Irish territory saw a penalty go Racing's way with five minutes to play.
Barkley made no mistake to hand the home side the lead once again, Bergamasco stretching that further with a late penalty after a Munster ruck infringement.
Last season's semi-finalist Edinburgh suffered a humiliating 45-0 home drubbing at the hands of Saracens.
The English club scored tries through Sam Tomkins, Owen Farrell, Chris Ashton, Alex Goode and Charlie Hodgson, the fly-half also scoring four penalties and four conversions for a personal haul of 25 points.
In Pool 3, Connacht saw off Italian club Zebre 19-10.
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