James Gray and Kal Naismith scored the goals for Stanley, who had to come from behind to triumph after Alex Henshall put the Pirates ahead.
Manager Beattie was left to reflect on them breaking their duck.
He said: "I thought we deserved it. I thought we were the better side.
"It feels good but as I've said all along I have faith in the squad and we don't carry ourselves like a side bottom of the league. We played some great stuff and they deserve it. I'm really pleased for the players.
"We've got our first win but nobody is going to get carried away. I've never had any doubts. Other people have had their doubts and been quick to point them out to me. Fortunately people that know people know what's going on here and it's not just a quick fix, it's for the long term."