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United lost Mark Durnan to a red card for felling Brian Graham in the Scottish Cup quarter-final.
"What a shift from our players, going down to 10 men the fighting spirit was excellent," said Paatelainen.
"Even though we were a man short we wanted an attacking threat, hence the 3-3-3 formation, and it worked."
Paatelainen's side are eight points adrift at the foot of the Premiership table, but have a game in hand over second-bottom Kilmarnock.
And the Tangerines now have a semi-final to look forward to next month after Henri Anier's double and Billy Mckay's headed winner defeated County.
"We know we are a match for any team when we play well," the Finn told BBC Scotland. "Today we did. That hasn't changed at all.
"Yes, we have a long way to go; we're nowhere near safety yet. The boys' attitude is tremendous on the training ground and in the matches as you can see, but hopefully, by hard work, I believe we can do it."
Ross County manager Jim McIntyre meanwhile bemoaned his side's failure to prevent Dundee United's deliveries into the box - from which they derived much of their attacking success.
"We've blown a golden opportunity to get to another national semi-final," he said. "Quite frankly we didn't stop crosses - their goals have come from three crosses and it's not good enough.
"We've got to get out, get close, but when United went down to 10 men we seemed to stand off them and allow crosses to come into the box. We've got to be much better than that.
"We've been masters of our own downfall. You've got to give United credit, of course, but we've not picked up our men and we've been punished."
County tackle Hibernian in next Sunday's League Cup final and McIntyre revealed midfielder Martin Woods "felt ill" at half-time, prompting his substitution.
But Jackson Irvine, who was also withdrawn, is not currently a concern for the showpiece at Hampden.
An "exhausted" Mixu Paatelainen praised the resilience of his Dundee United side after they twice came from behind to beat Ross County 3-2.