Barca were already guaranteed top spot in the pool ahead of Manchester City before kick-off, with Gladbach similarly secure in the Europa League qualification place at the expense of Celtic.
Luis Enrique retained just three players from his side that drew El Clasico on Sunday, but Lionel Messi started, two goals short of Real Madrid rival Cristiano Ronaldo's Champions League group-stage record haul.
Messi only managed to score one in a first half that saw most of Barca's chances fall his way, with Arda then at the double with a pair of quick-fire strikes after the interval before he secured himself the match ball midway through the second half by striking his third.
Arda had earlier set up Messi's goal in a redemptive outing for the winger, who had not started in this season's Champions League before Wednesday's tie.
Having little to lose did not prompt a cavalier approach from Gladbach, who operated with a six-man defence when Barcelona had the ball - which accounted for the majority of the match.
Arda played a pass behind the advancing Messi with Barca's first clear chance, but the Turkey international got things right when presented with another opportunity two minutes later.
Messi pounced on a loose ball to race through the centre before playing in Arda on the left and then darting into the box to sweep home the subsequent cut-back.
Yann Sommer had to parry Messi's fierce drive over the crossbar two minutes later as the Barcelona number 10 - playing in a withdrawn role behind Paco Alcacer - inspired some of the Catalans’ lesser lights.
Barca probed and prodded Gladbach, who offered little in return, and Messi was denied his landmark goal by Tobias Strobl's last-ditch tackle two minutes from the break after fleet-footed work from Aleix Vidal and Denis Suarez on the right.
That same wide-right pairing combined again five minutes after half-time, as Suarez provided Arda with a simple header and he latched onto a pass from Vidal to clip the ball home and put Barca in sight of another big home win, following 7-0 and 4-0 victories over Celtic and City already this term.
Alcacer did not take his chance to prove his first-team worth, blasting a one-on-one chance straight at Sommer, but the former Valencia striker did make Arda's hat-trick finish a simple one, playing a ball across the six-yard box that the winger netted.
Sommer produced a stunning one-handed save to deny Messi's chance to claim his place in the record books with a diving header, before Arda was afforded a richly deserved ovation when he was replaced by Champions League debutant Marc Cardona.
The celebratory mood gave way to a meandering finale, but a first win in four games - and such a dominant one - will put a spring back in Barca's step as they hunt down Madrid in LaLiga.