Rovers managed to nab Armstrong in a £1.75 million deal, with the player signing a four-year contract.
Armstrong spent the second half of last season on loan at the club, scoring 7 goals in 21 games.
Newcastle have now allowed him to move there permanently.
"I think he's a snip, a bargain really, the tender age he is," Mowbray told Sky Sports.
"When you look at some of the numbers going around for some players at the moment, I think £1.75m is a snip for such a young lad who has such power, dynamic speed, can score off both feet and be a real threat in behind.
"He's still got learning and developing to do, but I'm delighted he's here and delighted he's made this his home for the next three or four years.
"The challenge for Adam is to first of all get into our team in his preferred position, which is down the middle, but I've got great belief in him.
"I know he can score goals, I know the attributes he has, so we just have to provide the ammunition for him and I'm sure he'll finish it off."