The result means Britain came from two rubbers down for the first time in 83 years to set up an unlikely World Group play-off in September.
Evans completely outclassed Donskoy throughout, manoeuvring the Russian across the court with an array of shots befitting a player far above his world ranking of 325.
Earlier at Coventry’s Ricoh Arena, British number two James Ward had beaten Dmitry Tursunov 6-4 5-7 5-7 6-4 6-4 in an epic five-set clash to level the Davis Cup tie at 2-2 heading into the final rubber.
"Wardy deserved so much credit for the way he played earlier, even on Friday (where Ward lost in five sets to Donskoy after leading 2-0) it was a joke", an emotional Evans said on court afterwards.
"It's testament to the hard work all the boys put in," GB captain Leon Smith added.
"It just shows you that anything is possible. At the end of the day it was just two outstanding performances. "Before the tie I genuinely knew we could cause an upset. If you put all the ingredients in the mix then anything is possible. We certainly believed and on the practice courts we knew we could do it. "The atmosphere was fantastic too, we really enjoyed it in Coventry. For James to beat Tursunov in five after what he did on Friday was excellent. "And, of course, what Dan has come in and done fantastic. He stuck to his game plan and executed it perfectly."
Britain faced an uphill task after Evans and Ward lost their opening singles rubbers on Friday, but yesterday’s resounding doubles victory by Jonny Marray and Colin Fleming gave them a glimmer of hope heading into the final day.
Ward flew out of the blocks in Sunday’s first match and deservedly took the first set after a superb display of serving before failing to capitalise on numerous break opportunities in the second.
Tursunov responded by taking the set with his first break point of the match, and soon wrapped up the third but Ward found something extra and forced a final set.
The Brit broke his rival in the fifth game of the final set after the Russian double faulted and he never looked back as he sealed the win to take the Davis Cup tie into the decider.
Despite being under intense pressure, Evans looked composed as he opened his match with a flurry of well worked points.
And it was Evans who broke first, smashing a weak Donskoy second serve back past the Russian in the seventh game.
The two exchanged breaks at the start of the second set but, just as it looked as though the Brit might crumble, Evans seized the initiative once more with a wicked crosscourt forehand to break Donskoy once more.
The Brit did not look back, taking the second set 6-4 before racing into a 4-0 lead in the third as Donskoy’s game fell apart.
Although the Russian did manage to get on the scoreboard, he could not prevent Evans breaking him one final time to complete Great Britain’s remarkable comeback.
As the Russian’s final shot drifted into the tramlines, the British squad stormed the pitch to celebrate one of the great moments in British tennis.
It was a stunning comeback and, with Andy Murray to come back into the side, British tennis can look forward to more exciting times again.