Stanley keeper Marcus Bettinelli denied Plymouth target man Marvin Morgan one on one in the second minute and although he was beaten three minutes later, Rommy Boco's angled drive hit the outside of the post.
Striker Reuben Reid fired wide from 10 yards in the 25th minute and gave Bettinelli an easy save in the 38th minute when teed up to score by Hamza Bencherif's nod-down, but Stanley grew in stature as the game progressed to end a six-match losing run in League Two.
Beattie said: "The clean sheet is massive for the boys and I really enjoyed watching them today.
"We were lucky not to be two or three goals down in the opening 10 minutes but we were much more accomplished once we learnt to deal with their threat.
"We survived a little bit more pressure at the end of the first half but in the second half I thought the lads were magnificent and were unlucky not to win the game.
"It could have been more than a point, but we are not getting carried away. We will take a point and I am really proud of the players and the performance they put in today.
"They were tired out after the game and that's got to be a given every time we play. We need to put as much effort in as we can to win the game and we certainly did that today.
"We subdued the crowd and turned them a little bit. We are bottom of the league but we performed brilliantly. We created chances, we have shut the back door and we need to build on this point today.
"I have got no worries about this team and if we perform like we did today we will win more than we lose."