The pair scored in each half to give their side a deserved victory over the struggling North East side and go into next week's game against Manchester United with great confidence
A Barnett tap-in after a delightful move and a second-half header from Morison were enough to get the three points, despite Keiran Richardson's late consolation goal.
"It was a great win and think we deserved it, apart from the last ten minutes when we had our backs against the wall a little bit – especially going down to 10 men [after substitute James Vaughan went off injured]," Morison told Sky Sports.
"But it was a great result, you have to win back-to-back games in this league and that’s something we’ve managed to do."
When asked about opening his Premier League account he added: "It’s fantastic, I’ve managed to score at a lot of different levels. I scored an international goal a couple of weeks ago and I managed to get a Premier League one as well.
"It’s good times at the minute and long may it continue."
"You’re judged on your goals [as a striker] and the most difficult thing is they just look at strikers and think you should be scoring every week," Morison continued. "It’s a big step up to this level and it’s tough –you don’t get as many chances."
"Obviously I’ve missed a couple but I’ve managed to put one away tonight and it’s pushed us over the line in the end."
Norwich travel to league leaders Manchester United at the weekend and the striker believes his side can get a result there as they are brimming with confidence.
"We go there looking for another win and make it three games out of three," he added. "We wanted to get back-to-back wins and puts us up into a nice area in the table and we can go there with bags of confidence."
Fellow goalscorer Leon Barnett was pleased with the way his side went about their business.
"The boys done well to put in a shift and it starts from the front. I thought we defended well and we took it from there really," said Barnett.
The defender also scored his first Premier League goal and praised Morison for his contribution too.
He continued: "I'm happy to get the goal – it’s a bit of a bonus - and Stevie’s done well to get his goal but the main thing is the three points and we’re all happy."
When asked about whether he was surprised by the room Sunderland allowed, he said: "I don’t know if they’ve seen the movement!
"I just stayed in the box. It was from a corner and obviously I just stayed up and I was fortunate a good ball came in and I finished well, really.
Norwich have been unlucky so far this season, conceding a goal and giving away penalties in every game in the Premier League this season before tonight, with Barnett guilty of giving away two of them.
Nonetheless, he believes being unfortunate in the past few games has helped them.
"We don’t want to get carried away," he continued. "We should enjoy picking up the three points, and we did well against Bolton. In the weeks past we’ve been unfortunate. Obviously the penalties and not getting the right result but we’ve done well today good performance."