Goals from Tom Bradshaw and Conor Hourihane gave Barnsley a two-goal lead at the break in the Sky Bet Championship match.
The visitors produced an improved second-half performance, with Nelson Oliveira pulling a goal back, but they were unable to score an equaliser.
Bradshaw opened the scoring with a fine finish in the 12th minute, sweeping the ball home after meeting Josh Scowen's cross from the right.
The hosts increased their lead five minutes before half-time when Hourihane found the net with an impressive 20-yard effort.
The goal from Oliveira (52) was one to remember as he beat Adam Davies with a stunning low strike from distance.
Heckingbottom explained: "It was a case of two teams playing two different shapes and I thought we imposed our shape on them better in the first half.
"They made changes at half-time and they imposed their shape and system on us at the start of the second half.
"We restricted them to shots from distance and we were disappointed with the goal we did concede.
"It changes the mentality and we had to get to grips with the middle of the park again. They were much stronger with their midfield three so we had to change and regain control of the middle of the pitch.
REPORT: Barnsley 2 Norwich City 1 - Reds continue to impress and close gap on Canaries"We didn't create chance after chance, but I thought we scored two good goals.
"I don't think there's ever been a dip in confidence or belief, if I'm honest. We know it's going to be tough but that's the nature of the league.
"You need to not get down if things aren't going your way and don't get carried away if they are going your way."
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