Russia needed the win to stand a chance of going through after a 1-1 draw with England and a 2-1 defeat to Slovakia left them at the bottom of Group B.
Wales enjoyed a promising start in opening minute before Arsenal midfielder Aaron Ramsey put them ahead after 11 minutes of play.
Ramsey ran on to Joe Allen's superb pass that split the Russian defence and dinked the ball over the onrushing Akinfeev and in the net.
Neil Taylor doubled the advantage for Wales nine minutes later beating goalkeeper Akinfeev on his second attempt after a very smart pass from Gareth Bale.
In the remainder of the match, Welsh star Bale constantly threatened the Russian goal, forcing goalie Akinfeev to work hard for blocks in the 39th and 47th minute.
Bale had his third Euro goal seven minutes after the hour mark. Ramsey ran across the edge of the area and chipped a pass through to Bale, who found himself one-on-one with Akinfeev before coolly dinking the ball past the veteran keeper to seal a famous Welsh win.
The convincing victory fired Wales through to the knockout stage as Group B leaders with six points, while England was the second team to advance following their goalless draw with Slovakia in another game the same day.
With just one point after three matches, Russia became the third East European team to be out of the tournament, after Ukraine and Romania.
