Port Vale had the first clear chance of the match after 13 minutes through captain Marc Richards, the former Blackburn striker, who latched on to a cross but his diving header was saved easily by Shwan Jalal.
In the 34th minute pressure from the Bournemouth attack inside the Vale penalty box saw winger Molesley pick up a free ball. He went past Lee Collins and sent a driven effort towards the goal but a sea of bodies deflected the ball out for a corner.
From the resulting set-piece, veteran striker Steve Fletcher almost headed home from six yards, only to see the ball bounce wide.
The first half ended with the Cherries on the attack. Warren Cummings tricked the travelling defence with a low pass to the near post from a dead ball. Danny Hollands broke free off his marker and hit a left-foot shot wide of Joe Anyon's goal.
Anyon kept Vale in the game early in the second half with two fantastic saves. In the 52nd minute Molesley broke free from the right wing and fired a low shot at goal.
Anyon, who looked to have been wrong-footed, managed to claw the ball away to safety.
Seconds later Anyon again denied Molesely but this time with an acrobatic dive following a long-range pile-driver.
In the 81st minute, Bournemouth sub Liam Feeney sprayed the ball across the width of the pitch. Striker Brett Pitman tucked inside and let fly at goal but keeper Anyon was well placed and saved comfortably.
In the dying minutes Vale striker Richards had the chance to win the match, but blazed over from 20 yards, leaving both sides with one point and leaving Bournemouth in the relegation zone.