Will Hoskins netted his 19th goal of the season, and his third against the Magpies, to settle a game in which the Pirates controlled possession, if not chances.
The three points moved them off the bottom of the table, and Campbell said: "Everyone is absolutely elated with the manner of the victory.
"The scenario we're in is that every game we approach, we want to go and try and win it. If we give that effort, I know we've got the ability in the team to win more than enough games to get us out of the trouble we're in."
Hoskins took the headlines, but the game was won by an 85th-minute penalty save from goalkeeper Conrad Logan, who kept out Alan Gow's spot-kick after Febian Brandy had been brought down.
On the award of the penalty, Campbell said: "It was very, very soft - I don't even think there was a coming together. Their lad went down very, very easily. I think we got our just reward because I don't think it was a penalty and Conrad made the save for us."
Source: PA
Source: PA