After a low-key first half at Scotstoun the Warriors trailed by three, but 14 points from the fly-half's boot and a penalty try just after the restart earned Glasgow their seventh GUINNESS PRO12 win of the season.
Weir's side now face a mammoth December, with a European double header against last year's champions Toulon before home GUINNESS PRO12 clashes with table-topping Munster and Scottish rivals Edinburgh, and the 23-year-old admitted it felt good to get some momentum.
"It was the performance that we'd have hoped for," he told Glasgow Warriors TV. "The conditions were tough with a greasy ball and that meant for a lot of driving line-outs and keeping the ball a lot with the forwards.
"Credit to the pack, they got through a lot of work there.
"It was a tough game to try and get the momentum but we are delighted with the win. Certainly after the defeat last weekend it's a step in the right direction.
"That last kick from me was a little sneaky going off the top off the post, but kicking is part of the game and I do a lot of work on it. Thankfully we didn't miss out there and grabbed every point.
"We've got a big period ahead of us, where three of our next four games are at Scotstoun. It's great that we have the momentum behind us and a strong home record.
"We always enjoy playing here at our home ground and appreciate the fans always coming out.
"It's going to be a huge challenge but the club is ambitious so it's one we are certainly looking forward to."
Glasgow move up to third in the GUINNESS PRO12 thanks to the result, level on points with second-placed Ospreys while Munster's bonus points gives them the edge at the top.
Head coach Gregor Townsend added: "I'm very pleased with the four points as the Dragons are a very tough team to play against.
"I felt that we put in a huge effort in the second half, particularly at the end, to get the win so I'm very pleased with the players."
Click here for the best images from Glasgow's hard-fought win against Newport Gwent Dragons
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