The Europeans, who were silver medallists at the Beijing 2008 Games, were cruising to the points in Group A as they led 19-11 in the early exchanges of the second half, but a golden spell by the African champions saw them level at 23-23.
However, Russia regrouped and flexed their muscles to get over the finishing line by a three-goal margin, Liudmila Postnova top-scoring with five goals and Liudmila Bodnieva, Tatiana Khmyrova and Ekaterina Davydenko all grabbing four.
Angola's performance was underpinned by five goals from Isabel Guialo.
In the second match of the morning session, Republic of Korea comfortably saw off Spain 31-27 in Group B, which was something of a surprise result.
Eun Hee Ryu was chief destroyer for Korea with nine goals, while Carmen Martin Berenguer notched seven for the Spaniards.