Russia had the best score after one event in the team competition, posting a 28.375 in the five-ball routine. It had the highest artistic score of any team at 9.625.
Russia has won gold at the past three Olympics and finished 0.275 points up on second-place Italy, which has won each of the last three world titles.
Belarus is third in the group all-round standings, while Great Britain and Canada sit 11th and 12th, respectively.
Russia also saw two of its gymnasts grab the top spots in the individual all- around in Daria Dmitrieva and Evgeniya Kanaeva. Dmitrieva had the top score in the hoop event (29.000) and the best total score of 57.800, while Kanaeva's 29.525 mark in the ball led the event and helped her to a second-place overall total of 57.625.
Kanaeva, a three-time world champion, is trying to become the first rhythmic gymnast to win two Olympic gold medals in the all-around after also taking the title at the 2008 Beijing Games.
Belarus' Liubou Charkashyna is third with a 56.450 score.
Great Britain's Francesca Jones is 21st out of the 24 gymnasts, just ahead of American Julie Zetlin.
On Friday, the individual gymnasts will perform the clubs and ribbons routines, while the groups perform their second of two routines with three ribbons and two hoops. The top 10 individual athletes, as well as the eight highest-scoring teams, advance to the final, to be held this weekend.