Dan Parks turned in a superb kicking display in dreadful conditions, slotting home four penalties to secure the win.
The Warriors will face more Welsh opposition on Friday at Firhill when the big-spending Ospreys come to town and Al wants his team to build on the momentum they gained in the Rodney Parade win.
The Warriors' skipper told the Evening Times: "Despite the conditions being horrific Dan was superb. He really controlled the game for us and hopefully he can carry that form on against the Ospreys on Friday at Firhill.
"We knew we would have to front up against the Dragons and we did that. Defensively we were superb and it is so important to open with a win and a decent performance.
"But we now have a platform for the visit of the Ospreys and it is up to us to make the most of that."
Warriors' coach Sean Lineen agreed that Glasgow's strong defence had laid the foundations for the win.
Sean said: "Our defence was absolutely superb. We also showed a lot of spirit and really hung in there and these are all qualities that are going to be very important over the coming season.
"But now we have more Welsh opposition in the Ospreys on Friday at our place and it is vital we use that to build some real momentum."