Hope Akpan and Joe Walsh fired Town into a commanding lead before Lee Barnard reduced the arrears, but Nicky Adams sealed Crawley's second successive win midway through the second half.
Barker said: "We sat with a game plan to win the ball in certain areas and we are almost set-up to be a counter-attacking team.
"Our last 10 goals have been scored by non-strikers and that's a stat I like."
However, victory came at a cost and Crawley are set to be without defender Claude Davis after he was forced off at half-time with a recurrence of a hip injury.
Barker added: "We're rapidly running out of central defenders and this is a blow as Davis was earlier out for three weeks with a similar injury."