It wasn't necessarily non-violent. Civil war started in Tajikistan almost immediately, the Nagorno-Karabakh War between Armenia and Azerbaijan had been going on since 1988, Moldova had a brief war with Transnistria, and Georgia saw fighting between its separatist regions. Thousands died in these combined conflicts, and both Armenia and Azerbaijan are still shooting at each other today, while Georgia most recently fought Russia in 2008 over its regions, and the current Ukraine conflict could be argued as a legacy of this. But that's all too recent for here.

Now, I imagine you mean why didn't Russia engage in civil war or any major conflict. Well even there they had the 1993 shelling of the White House; that total conflict saw at least a couple hundred killed, possibly thousands. But in terms of 1991, I'd say that because the USSR was a federation, the 15 constituent republics (or 12 by that point, as the Baltics had effectively split off by then) were more concerned with their own future, and their problems are noted above.

Russia itself though was in a position to avoid conflict. Gorbachev worked until nearly the dissolution of the USSR to preserve it, using diplomacy and negotiation, as he was not widely supported in the military. This is evident by the coup attempt in August 1991, where hardliners within the government tried to overthrow him and preserve the USSR. That was probably the closest Russia got to an actual civil war, and indeed there were plans to fire on the crowds that had gathered to stop the tanks. Had that gone through, it would have been a massacre and more than likely resulted in a war. But fortunately common sense prevailed, and the coup was ended without any deaths.

Note also that I don't include Chechnya here, because the first Russian invasion only happened in 1994; prior to that Chechnya had been largely de facto independent, but left to its own by Russia. Though Russian forces were sent to Grozny in November 1991, they quickly retreated in the face of Chechen forces, and no shots were fired at that point.

Edit:

For anyone interested in a clear, simple look at the dissolution of the USSR, I'd suggest Moscow, December 25, 1991: The Last Day of the Soviet Union by Connor O'Clery. It gives a look at the events leading up to Gorbachevs resignation, interspaced with a full look at the date in question. Its a short book (around 300 pages) and a quick and easy read that will give anyone unfamiliar with the era a decent understanding of what happened.