First-half goals from Nicky Adams and Billy Clarke and a late penalty from skipper Gary Alexander gave the Red Devils a thoroughly convincing 3-0 home victory over Blue Square Bet South opponents Chelmsford City on Saturday.
The visitors' chances of springing an upset were hit by the 35th-minute sending-off of striker Donovan Simmonds, but such was Crawley's dominance that home goalkeeper Paul Jones was not tested once during the whole match.
Barker praised a "professional" attitude and, when asked who he would like to face in the next round, replied that either Arsenal or Manchester United away would be his preferred choice.
"I'm a big fan of (Arsenal manager) Arsene Wenger so maybe a trip to the Emirates would be nice," he said. "I'm also a big admirer of Sir Alex Ferguson and facing Manchester United at Old Trafford would also suit us."
Barker was delighted that there was no complacency whatsoever shown by his players against the non-league side and added: "The way we fizzed the ball around was excellent.
"There are three divisions between us and Chelmsford so it was always going to be tough for them but our attitude and professionalism stood out.
"We had gone six league games without a win going into the tie, so hopefully the mood around the club will change and become a bit more confident."
While focusing his hopes on a big away draw, Barker is mindful of other potentially difficult ties and added: "There is still a few banana skins left in the competition.
"All in all we are pleased at the way things are going, to be only three or four points off second place in League One shows how well we've done."