The 28-year-old former Germany midfielder, who joined this season from Bayer Leverkusen and initially struggled for form, delivered a standout performance in the first half.
"I feel vindicated for our patience with him," Dortmund coach Thomas Tuchel told reporters. "Gonzo always remained calm."
"We are pleased with the win, the many chances and the 70,000-plus spectators."
Paderborn, with former Bayern Munich captain and rookie coach Stefan Effenberg in the dugout, took a surprise lead when Dortmund keeper Roman Buerki attempted to dribble past Srdjan Lakic only to see the Croatian slide in and score in the 21st minute.
The hosts, second in the Bundesliga, came close to an equaliser two minutes later as Shinji Kagawa scraped the bar with a powerful shot. Adrian Ramos then did even better, latching on to a Castro pass, sprinting clear and rounding the keeper in the 25th minute.
Ramos turned provider five minutes later, flicking a high ball for Castro to make it 2-1 before Kagawa made amends for his earlier miss after another Castro assist in the 43rd.
Two quick goals from Ilkay Guendogan, including a penalty, early in the second half made sure of the win and late goals from Lukasz Piszczek and Henrikh Mkhitaryan completed the rout.
Borussia Moenchengladbach won 2-0 at Schalke 04, their second victory over the Ruhr Valley club in four days following a 3-1 league victory marred by a brutal challenge by Johannes Geis on Andre Hahn.
Despite missing striker Hahn, Gladbach continued their winning domestic run under new coach Andre Schubert. They have triumphed in five straight league games since he took over.
Lars Stindl scored against the run of play after Schalke's Joel Matip slipped outside the box and the hosts then missed several chances before the break.
A Thorgan Hazard penalty in the 53rd minute took the wind out of Schalke's sails and they rarely threatened in the second half.
In the other all-Bundesliga clash, Werder Bremen edged out Cologne 1-0 with Anthony Ujah netting against his former club.
Bayer Leverkusen also had few problems against fourth-tier Viktoria Cologne, easing to a 6-0 victory, while Augsburg, the Bundesliga's bottom club, defeated Freiburg 3-0.
Bayern Munich beat holders VfL Wolfsburg 3-1 on Tuesday.
(Reporting by Karolos Grohmann; Editing by Tony Jimenez)
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