The Welsh dragons got off to a flying start with Ramsey and Taylor scoring within the first 20 minutes, they continued to dominate for the rest of the period with good chances from Sam Vokes, Ramsey and Bale, but failed to increase the score-line any further.
The second period didn't lead with the same ferocity, but there were good chances for Bale and Denis Glushakov, but couldn't add to the scoreline until Bale added a third to secure their passage into the last 16.
Chris Coleman's side went into this match knowing that a win or even draw, could be enough to see them through the group stage, and almost had the lead within the first 60 seconds. Bale had been one of two Welsh goal scorers so far in the tournament, and almost added a third when he hit a thunderous effort, but Igor Akinfeev produced a excellent save.
However, ten minutes after that excellent opportunity, they managed to take the lead. Joe Allen produced an excellent through-ball from around the half-way point, it split the poor Russian defence right open and Ramsey managed to keep his cool against a rushing Akinfeev to chip it over and into the net.
Coleman, and the Welsh nation wouldn't have expected to see them add a second on the 20-minute mark. Bale had proved a havoc in the opening stages, and was at it again as he made a bursting run into the Russian half, and his deflected pass was perfect for Taylor who had all the time in the world coming in from the outside. The full-back nearly fluffed his lines, as his initial effort deflected off Akinfeev, but was alive to the situation to fire home at the second time of asking.
It almost came crashing down in the 26th minute, when a big kick from Akinfeev was misread by Ashley Williams, it allowed the ball to fall to Aleksandr Kokorin, but Wayne Hennessey did well to save the effort.
Looking for a third
Wales were looking to finish the game off as the half was coming to a close, and had some excellent opportunities to add their third goal in quick succession. Bale was at it once again as another great run allowed the ball to go through to Vokes, but it was saved by Akinfeev, Bale followed that up with a looping header, but the flag was raised for offside.
Ramsey followed that up with their last real opportunity of the first period, Ramsey was on a high and looking for his second, the Arsenal midfielder tried one from distance and forced Akinfeev to produce an unorthodox stop.
Slowing the pace
Coleman's side didn't have the same intensity as the first, but still dominated proceedings.
Bale had another excellent opportunity as he ran onto the pass from Ramsey, and was looking to beat Akinfeev from the tight angle, but the deflection from the foot of the Russian took it wide.
Leonid Slutsky's side had barely been involved attacking-wise in the contest, but had a rare opportunity when Dimitri Kombarov 's cross was deflected to Glushakov but the half-volley flashed past the post.
Into the last-16
Any nerves before the game had been extinguished by Wales' excellent play, and their historic passage was sealed with their third goal.
Ramsey and Bale combined once again, with the through-ball from Ramsey, left by the offside Chris Gunter, to run through for Bale and he made no mistake as his effort with the outside of his boot was deflected in.
Russia had an excellent opportunity to get a consolation, Pavel Mamaev put an excellent ball across the face of goal, which found Artyom Dzyuba, but he managed to put it over from little over six yards which summed up their sides performance, and Wales' dominance.