The home side twice surrendered the lead against their fierce rivals in an enthralling encounter at the boisterous La Bombonera in Buenos Aires.
The much-anticipated clash, the first between Argentina's biggest clubs in the final of South America's top club competition, was delayed until Monday (AEDT) due to heavy rain in the capital.
Boca went ahead through Ramon Abila only to see Lucas Pratto level practically from kick-off, and its advantage was again snatched from its grasp when Izquierdoz's wayward header cancelled out Dario Benedetto's goal.
It leaves the tie tantalisingly poised ahead of the second leg in 13 days.
It was River which threatened first, Agustin Rossi keeping out Gonzalo Martinez's free-kick before Lucas Martinez Quarta made a mess of a free header from the resulting corner.
Rossi them denied Santos Borre with a fine one-handed stop, and Boca's difficult start got worse when Cristian Pavon was forced off after 27 minutes with what looked like a hamstring injury.
The Bombonera was rocking 34 minutes in, though. Abila saw a shot saved by Franco Armani, but fired the rebound through the grasp of the keeper and in.
The lead lasted barely 60 seconds, with Pratto racing on to Martinez's pass and firing across goal and into the bottom-right corner.
Another chance for Martinez came and went before Boca snatched the lead again seconds before half-time, Benedetto glancing Sebastian Villa's deep free-kick high into the net.
A similar set-piece routine almost created a third for the home side, only for River to equalise again in similar fashion, when Izquierdoz headed in past the helpless Rossi from six yards out.
Boca introduced Carlos Tevez from the bench and the veteran striker almost provided the winner, first lashing a shot just wide of the left-hand post before teeing up Benedetto to shoot straight at Armani with the goal at his mercy.