Just seconds in, Russia goalkeeper Igor Akinfeev was struck in the back of the head by a flare thrown from the stands at Stadion pod Goricom.
While the teams played on, a second incident during the second half led to the abandonment.
Roman Shirokov had a penalty saved by Montenegro goalkeeper Vukasin Poleksic in the 66th minute.
The ball went out for a throw-in, which Dmitri Kombarov went over to take – only to refuse to when he appeared to be struck by an object thrown from the stands.
That led to several scuffles on and off the pitch, with the encounter called off after 67 minutes when the score was 0-0 as Russia left the pitch.
Earlier, Akinfeev was stretchered from the ground and his injury led to a 30-minute delay.
The 28-year-old was reportedly taken to hospital with concussion and burns on his neck.
Russia and Montenegro sit third and fourth respectively in Group G, with the result of Friday's clash yet to be decided.
Euro 2016 qualifying was marred by an incident last year, when a drone carrying a political banner interrupted Serbia's clash against Albania.
Both teams were punished over the incident.
