Holdsworth's men had not won a game in all competitions since January 12 but dominated their visitors from start to finish, although it took an 83rd-minute scramble to seal the win.
The win lifts Aldershot up to 20th in the table and just three points behind the Gulls, leaving Holdsworth to praise the spirit of his players.
"We were well worthy of that," he said. "I thought the players were excellent, their appetite and their desire to win the game was great.
"You could see that from the goal, Troy Brown threw is whole body at it and you saw what it meant to everybody, the players and the fans, the whole group together.
"It was a fantastic feeling when that final whistle went. We were resolute and the players know it. We looked solid.
"It emulated everything that the players have got in the dressing room. There's a togetherness in there, the team spirit is good, the squad is good.
"We now need to pick them up again and get them healed and rested for another good challenge against Fleetwood on Saturday.
"Every player gave everything so I'm proud of all of them but there's no time to dwell on it is there. We've got another one on Saturday and we go again."
- Sports & Recreation