Hope Akpan and Nicky Adams gave the League One newboys the lead on separate occasions in south London, before Darren Ward and Shaun Batt levelled things up for the hosts.
The match ended 2-2 and, after extra time, went to penalties, where debutant Paul Jones proved the Reds' hero by saving penalties from Darius Henderson and Jack Smith.
It was recently-appointed manager Barker's first match at the helm and he believes the win over the Championship outfit was fair.
"I thought it was a deserved victory in the end," he said.
"We deserved to win it in normal time and probably almost in extra time, to be fair.
"Sometimes you get a bit of justice in penalty shootouts and we did in the end.
"[To have a goal disallowed in the second half but carry on] just shows the character of the players.
"They didn't stop doing what we had done all night, which was pass the ball around and work extremely hard and be disciplined in our shape.
"We did all of that and to do that for two hours away at a Championship side speaks volumes for the players."
Crawley's attention now turns to Saturday's match against Scunthorpe - the club's first ever in the third tier.
"People will go into that on a real high now and I think cup runs keep enthusiasm going," Barker added.
"Obviously Crawley have had some fairly famous ones over the last couple of years so who knows what could come out in the next game.
"It'll certainly keep people in good sprits but now we have a different agenda over the next few days and that is to prepare for Scunthorpe, who I believe went into extra time as well after 5-5 thriller.
"I am quite pleased about that as it could have been quite detrimental to us otherwise."
Defeat after 120 minutes of football was hardly the preparation Millwall would have been hoping for heading into their Championship opener against Blackpool at the weekend.
However, manager Kenny Jackett was keen to draw on the positives after a difficult loss on home turf.
"We did well to come back twice," he said. "Overall I thought that a draw was about fair, about the right result.
"Both sides had chances but we've hit the target with the two penalties and the goalkeeper has made two very good saves - particularly the one from Smith was an excellent save.
"Batt scoring will hopefully break a big barrier for Shaun.
"He has battled very hard, it has been a long road for him and to get the goal hopefully lifts him."