And it could have been costly had Fa'atau not knocked on as he headed towards the line again.
Gregor did extend Glasgow's lead to 10 points midway through the first half, but Arlidge almost immediately cut that back with a successful kick himself, giving the hosts some impetus.
They searched for a try but had to be content with Arlidge chipping another three points off the Warriors lead in front of the posts.
But they got their try just before the break after Glasgow were reduced to 14 men when Kelly Brown was yellow-carded for deliberately killing the ball.
From the driving line-out, patience paid off and Gareth Wyatt went over in the corner to make it 11-10 after an unsuccessful conversion.
Glasgow escaped giving away another try when Luke Charteris knocked on but Arlidge's drop goal on the stroke of half time gave the Dragons a 14-10 lead.
The New Zealander added another three points at the start of the second half with a penalty.
But Glasgow were not ready to give up and on 53 minutes, Max Evans got the try on the right-hand side after slipping under the grasp of Joe Bearman with Gregor levelling with his conversion.
They were level for just two minutes though as the Dragons were back in front when Wyatt made the most of a fortuitous bounce to go over in the corner for his 19th Heineken Cup try.
And Colin Charvis, making his first start of the season, extended that lead with 13 minutes to go by driving over from a line-out to make it 29-17 with Arlidge's conversion.
The Dragons could not kill off the game and allowed Glasgow back in when Alastair Kellock went over, only for Gregor to miss the conversion in front of the posts.
Arlidge made sure there would be no late comeback by the visitors when he kicked another drop goal with three minutes to go.
Dragons coach Paul Turner: "A win is a win, but I was disappointed not to get the bonus point.
"But at 10-0 down when we seemed to be doing all of the playing, it was good to re-group and build an innings which is what we did.
"Our next move really is when you go two scores up, you put your foot down and go for that bonus point. It was never to be today.
"I was pleased with aspects of our play and Bath next week is a big challenge for us."
Dragons: Dolman, Wyatt, Sidey, Stcherbina, Fussell, Arlidge, W. Evans, Black, Willis, R. Thomas, A. Jones, Charteris, Bearman, Charvis, L. Evans. Replacements: Smith for Sidey (73), Coundley for Black (76), Hall for Charteris (71), Webb for L. Evans (73). Not Used: S. Jones, R. Lewis, Connor.
Glasgow: Stortoni, Faatau, M. Evans, Morrison, O'Hare, Gregor, McMillan, Va'a, Hall, Low, Palepoi, Kellock, Brown, Barclay, Beattie. Replacements: Henderson for Morrison (73), Jackson for McMillan (58), Tckachuk for Va'a (58), Kalman for Low (40), Barker for Palepoi (40), Swindall for Brown (63). Not Used: Vernon.
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