United were deservedly beaten through an own goal, when Andy Crosby blasted the ball off team-mate Marcus Williams, and a Kayode Odejayi strike.
"I thought the first half was an even encounter and the complexion of the game changed when they got the goal before half-time," said Adkins.
"We're bitterly disappointed. We never looked like we were going to create. We never got the ball out to Kevan Hurst. We got about three crosses in from the left-hand side throughout the whole game which is uncharacteristic of ourselves. But credit to Barnsley because they've stopped the supply.
"We never created as much as we normally do. It's the first time we've not scored a goal in nine or 10 months, so from our point of view we're bitterly disappointed."
Tykes boss Simon Davey admitted his side rode their luck in the opening stages.
"We were lucky to be in at half-time 1-0 up but you make your own luck," he said.
"It was scrappy and we didn't get going, but we got a bit of luck, I thought it was a better second-half. I thought we were in control of the game - we could've had a few more goals.
"We've worked very hard ensuring we work as units and defend as a team and it didn't look like our goal was going to be threatened - I don't think (keeper) Heinz Muller had a save to make."