Adam Reach scored the game's only goal after a sumptuous assist from Barry Bannan.
Reach's seventh strike of the season proved enough as Wednesday defended solidly to secure back-to-back clean sheets for the first time this season.
Afterwards, Bullen said that his players' decisiveness and willingness to go for the harder pass meant they deserved the victory on Teesside.
"When you're going through a hard run of results, it's totally human that confidence drops a little bit," Bullen said.
"You also tend to opt for the easier option rather than being brave and making that forward pass.
"Barry spotted that run from Adam.
"Adam makes runs like that week in, week out and 99 per cent of the time doesn't get the pass.
"But today he's taken it and it was a helluva finish.
"Luckily we showed discipline and professionalism to hold them out."
The Owls' hectic festive schedule continues with another away game at West Bromwich Albion on Saturday.
Lucas Joao and Josh Onomah both went off injured at the Riverside and Bullen said that he will assess the duo ahead of the weekend trip to the Midlands.