The draw moved the Reds three points clear of the relegation zone, although they remain 10 games at Oakwell without a win.
Norwich may also feel slightly frustrated, having drawn for the sixth time in a seven-match winless run.
On-loan Swansea striker Oli McBurnie broke the deadlock with his sixth goal since arriving in January but Josh Murphy broke his recent barren run to level.
After rarely having a sighter, McBurnie justified his inclusion in the Scotland squad as he flicked Andy Yiadom’s early cross past Angus Gunn in the Norwich goal moments before half-time.
Murphy come close with a long-distance shot before he pounced in the 71st minute, smashing the equaliser in after James Husband sent a low cross into the forward.
“I would say that I saw very good things, I had a very good feeling about the game,” said Barnsley boss Morais.
“The men (Barnsley team), they had moments of really good football and I’m proud of the quality that we started to show in such a short time, with so many games and not a lot of time to really work in the way we want and I think that we need.
“The boys they are doing well. They did well tonight. Points are always important. I used to say that when it is not possible to win all three, to pick the points that you can get, we pick one point and I think my men deserved more than the point they could pick-up tonight.
“I think we had the opportunities to kill the game before they levelled the game but we couldn’t.”