England flanker Phil Dowson grabbed a try in the second half for the Saints, with wing Jamie Elliott getting another and stand-off Ryan Lamb booting 20 points as the Dragons fought bravely for an hour.
Home wing Tonderai Chavhanga got only the third try hat-trick in Dragons history but that was not enough as defeat to Bath last week and the loss to Northampton means they are out of the competition.
Mallinder said: "I am relieved and pleased with that result, really.
"It is always tough to come down to Wales and the Dragons were very competitive and pushed us all the way. It was a good first half and we created a lot of chances.
"The Dragons defended well but we got on top in the second half and controlled the game. We put their forwards under a lot of pressure and eventually they lost someone to the sin-bin.
"The maul went well and the scrum also went well. Ryan Lamb was good and kicked well to the corners for us to play in the right positions."
It was Dowson's 100th game for the Saints and Mallinder said: "He is back and forth with England on Monday and Tuesdays and then back with us on Wednesdays so it can't be easy for him."
Home coach Darren Edwards took positives from his side's performance despite a seventh loss in a row for the Dragons.
He said: "We couldn't get our hands on the ball at times which was a shame for us because, when we did, we looked very dangerous.
"We are disappointed but there are positives to come out of this. We fronted up to the stiff challenge and I was happy with that."
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