Stanley took the lead after just nine minutes when Youl Mawene was judged to have committed a foul as the Cod Army defended a corner and Beattie scored for the first time in 54 matches from the spot.
The visitors doubled their advantage eight minutes later when Beattie beat Dean Howell and his shot squirmed under Town keeper Scott Davies before being nudged over the line by George Miller.
Barry Ferguson hit the post for Fleetwood from a free-kick before Stanley stretched their lead.
Beattie left Fleetwood skipper Steve McNulty in his wake and although Scott Davies was able to save his effort the keeper was unable to deny Rommy Boco lurking at the back post.
Jamie McGuire grabbed one back for the home side, stabbing the ball home at the back post from a Barry Nicholson corner and Junior Brown's overhead kick, on the stroke of 90 minutes, could have set up a grandstand finish but Ian Dunbavin was equal to the challenge.
