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The battle between the two teams was always close in terms of territory and possession, but Newport had more of a cutting edge in the difficult conditions and the victory will be some consolation following their early Heineken Cup exit last week.
Glasgow’s pack won good lineout ball and put pressure on the Dragons in the scrum, but found the home defence too strong.
The Dragons took the lead after 15 minutes when Gough won a lineout and drove to the line before popping the ball onto outside-half Ceri Sweeney to cut through to score and convert.
Eight minutes later South African full-back Percy Montgomery returned a weak free-kick to touch by Glasgow outside-half Dan Parks to the Glasgow line. Newport won the lineout and drove prop Chris Anthony over for Sweeney to convert.
In the 26th minute Parks reduced the deficit with a penalty when the Dragons were caught offside in their 22. Inside a minute of the restart he had also added a dropped goal.
There continued to be little between the sides with many errors in the difficult conditions, but the Dragons’ superior three-quarters kept them in front. Recalled Wales squad lock Ian Gough also had another big game for the Dragons and hooker Steve Jones was a livewire.
In the 53rd minute Wales international Hal Luscombe broke clean through the centre and put Montgomery over for a try, Sweeney again converting.
But Glasgow were finally rewarded for their effort in the 75th minute when Canadian prop Kevin Tkachuk was driven over for a try which Parks converted.
Parks added a late penalty to earn his side a deserved bonus point.