The County skipper was among the scorers as they racked up a 6-2 victory at Global Energy Stadium, and felt the manner of the win spoke volumes about the squad.
"We dominated the game early on and looked very dangerous," said Quinn
"We were comfortable at the other end as well.
"We got ourselves in front, and although they pegged us back, we showed the right manner to bounce back again scoring another three goals before half-time
It showed the character of the squad.
"When you are in a cup tie there is only one objective and that is to be in the next round, so we are delighted and happy with the performance."
United's Tony Andreu had cancelled out a sensational opener from Christopher Routis, but Quinn restored the home side's lead and further goals from Tim Chow and Liam Boyce put County 4-1 up.
Andreu scored again in first half stoppage time to keep the Arabs in the tie but late goals from Routis and debutant Jim O'Brien, signed on loan from Shrewsbury, rounded off a fine afternoon's work.
Quinn was delighted the side converted most of the chances that came their way.
He said: "There is one thing about us, we always create chances, that's never changed.
"We've got players who know how to put the ball in the back of the net
Boycie is on fire and the midfielders are adding goals - that's what happens when you are in a good bit of form, and that's what we're in at the moment.
"But we are not getting carried away
It's positive knowing you are going into games and you are going to create chances and cause the opposition problems.
"We showed that throughout the game and when we had to dig in and survive some pressure we were disciplined
I thought it was an all-round perfect performance."
Dundee United manager Ray McKinnon was unhappy to see his side concede so many goals, especially as they had threatened a second-half comeback with a number of chances.
He said: "For a team that had recently gone six or seven games without losing a goal that was disappointing.
"But there was some positives to take from the game
The second half performance, for the first 30 minutes, was good
On another day that could have been a slightly different game."
Source : PA