Dodgson, signed on loan from Burnley on summer deadline-day, put the Reds in front on 34 minutes with Cosgrove involved in the build-up by winning a flick-on from Mael de Gevigney's throw-in.
Substitute Aiden Marsh made the game safe in stoppage-time on a night when Jamie McCart also made his bow and captain-for-the-night Conor McCarthy made his return from injury after almost a year out with an ACL injury.
Collins said: "To win is good. There were more individual things that were probably better than the group. But that was only to be expected with the team we put out. It was a strong team, but with the individual situations (with debuts), it probably showed why it was a little disjointed at times.
"But ultimately, there were more positives.
"You look at someone like Sam. To be honest, Aiden Marsh probably deserved to start tonight based on his performances.
"But for the long term, it was important for Sam to get minutes. He had a difficult pre-season at Birmingham, so that was important.
"I thought Sam showed some of the things he will bring - just his size and physicality.
"Owen showed some really good attributes that will give us really good competition on the left side. For those two to make their debuts was great."
On Marsh's second successive goal in cup competitions at Oakwell, he continued: "Between him and Fabio Jalo, we have got a couple of really good young attacking strikers. Fabio would have featured, but has been on international (duty).
"Everything Marshy has done in the last few games, he has done really well. There's nothing better for a striker than putting the ball in the net."
On his side's third consecutive 2-0 win and starting off the EFL Group campaign with a win and the spin-off benefits of a potential run in the competition, he added: "Being able to progress gives us the chance for more opportunities. We've got a lot of competition in the squad and a lot of young players who need to play at this level and from that aspect, it's good.
"It was a good night to see these players who will be knocking on my door in the next few weeks and wanting to play.
"This is the best chance for them to do their talking.