The Russians were 2-0 up and cruising after Kovalchuk's ninth-minute goal, but three quick-fire goals, including two in a 15-second burst, put Austria in front before Russia leveled again at the end of the first period.
Alexander Radulov put Russia ahead again only for the dogged Austrians to pull them back through a Florian Iberer goal. After that the Austrian resistance crumbled and the Russians fired four more goals to complete the rout.
By winning their final group game in fine style, defending champions Russia will progress to the quarter-finals as no worse than third seeds. Austria are condemned to finish bottom of their group and drop from the elite to division one of world ice hockey.
In the early game in the Finnish capital, Latvia beat France 3-1 in a result that moved them out of the relegation zone and dented France's hopes of making the quarter-finals.
A third-quarter Brenden Dillon strike in Stockholm helped Canada edge Slovenia 4-3 in their final group game to move to the top of their group ahead of Switzerland.
Slovenia failed to win any games at the world championships and are relegated to division one for next year.
Earlier in the day, Denmark held on to beat Belarus 3-2, despite the latter fighting back from 3-0 down. Second-placed Switzerland can still win the group if they make it seven wins out of seven by beating Belarus on Tuesday.
Third-placed Sweden have also made the quarter-finals, with the final berth in the Stockholm group going to the winners of the crunch game between Norway and the Czech Republic.
In Helsinki, the United States can top their group with a win over Slovakia, with Russia and Finland also already qualified for the quarter-finals.
Slovakia, Latvia, France and Germany are all in with a shout of making the final quarter-final spot, with the latter pair due to face one another in a must-win match.
Should Slovakia upset the United States, they will go through, while Latvia will be looking to beat Finland to have any chance of progressing.