Rubin Kazan beat FK Krasnodar 2-0 to finish sixth and claim the last spot in next season's Europa League, while Dynamo dropped to seventh following a scrappy performance and had coach Dan Petrescu sent off in the first half after he stormed onto the pitch to confront the referee.
"I want our fans to know that I feel sorry for my outburst," the former Romania and Chelsea defender told reporters.
"We knew we needed to win today but came up short at the end as we missed our chances. We also missed key players today," Petrescu said in reference to Russia striker Alexander Kokorin and Hungary midfielder Balazs Dzsudzsak, who were both suspended following last week's brawl against Alania Vladikavkaz.
It was the third red card shown to the fiery Romanian coach in the past 12 months. He was banned for six matches after being ejected for pushing the referee late last season while coaching Kuban Krasnodar against Dynamo.
Petrescu, who was appointed Dynamo coach last August, was also sent off in a Russian Cup match earlier in the season.
Volga also missed a penalty in the second half but the point was enough to keep them in the top flight as they finished 12th.
Elsewhere, 13th-placed Rostov stunned newly-crowned champions CSKA Moscow 3-0 but now face a relegation playoff, while Zenit St Petersburg, who finished second this term after winning the title for the past two seasons, were held to a scoreless draw at Amkar Perm.
Big spenders Anzhi Makhachkala, who sealed third place - their best finish in Russia's top flight - with a game to spare last week, slumped to a 1-0 defeat at Kuban Krasnodar.
The win lifted Kuban into fifth spot and guaranteed them a place in Europe for the first time in the club's history.
"Obviously this game meant a lot more for Kuban, who were much more motivated than our team," Anzhi coach Guus Hiddink told Russian television NTV Plus. "I think today we lacked a bit of pace and determination as we already achieved third place."
Anzhi will have a chance to win their first major trophy when they take on CSKA in the Russian Cup final in the Chechen capital Grozny next Saturday.