Andrew Crofts scored his seventh and eighth goals of the season with a glancing header and a stunning volley in the first half.
Marc Tierney was allowed the freedom of Oakwell to switch the ball to David Fox who clipped the ball onto Crofts’ head for him to glance an opener with his forehead.
Things went from bad for worse for Mark Robins, managing against his former club, when fellow former Canary Jason Shackell cut his lip in a clash of heads.
As he received treatment on the sidelines, the extra space afforded to the Canaries resulted in Crofts’ second goal.
The visitors were unlucky not to score a third and add extra gloss to the victory when Steele fumbled a loose ball in his area on 80 minutes after a Grant Holt snapshot.
The Tykes lacked cutting edge in the final third throughout and fell to their first defeat in four home matches as the final whistle sounded.
Barnsley boss Mark Robins admitted: “It was a bad day at the office. We didn’t perform and we made it easy for Norwich.
“We didn’t start the afternoon off brightly enough and we didn’t cope with the conditions as well as they did. We conceded two goals that were poor from our point of view.”