Fly half Arwel Robson and wing Dafydd Howells scored crucial first half tries as the Dragons controlled territory and possession in wet and blustery weather.
“We know how hard it is to get a win in this competition so we are pleased,” said Jackman.
“Our defence looked much more comfortable and to only concede five points against a team like Zebre, who like to offload and run from their own half, is good.
“We felt quite comfortable and we made a step forward. We closed the game out comfortably too and I’m happy with four points.
“There are elements of the performance that we are frustrated by. We didn’t convert some opportunities in their 22 and lost some cohesion at the attacking ruck and got turned over a few times.
“But our scrum was rock solid and our kicking game was good and we dominated territory. In the conditions that was smart.
“I know people said we had an easy run, but the Kings beat Glasgow today and we certainly feel that they are on the up. Benetton Rugby and Zebre are more than competitive if you let them play.
“This team is still growing and we’re reasonably content, although we still have a lot of work to do.”
Dragons now travel to Glasgow Warriors next Saturday evening and Jackman is keen for his team to show they are taking steps forward.
“Glasgow are an excellent side and topped the league last year in terms of points,” he said.
“We saw the result with Kings and followed it. We will have a look at the game tonight, but we know we are going to a 4G pitch and it will be a fast tempo game.
“It will show us what we learnt from Leinster because they like to have a lot of possession. Then we have the Blues to finish off this block, they are coming thick and heavy.
“If you start panicking because of outside criticism or people trying to doubt you then you will stray away from your philosophy,” he added.
“We wouldn’t have panicked if we had lost today, but it makes life easier when you win. The boys will have a smile on their face on Monday morning.
“We know there are things to fix, but it is easier on the back of four points.”
Dragons return to action at Rodney Parade against Cardiff Blues on Saturday 6 October, 5.15pm. Early bird tickets are available until midnight Friday.
CLICK HERE to secure your tickets for the game.