Both sides missed chances but both goalkeepers had to be on their toes with superb saves in an entertaining game.
Wrexham could claim they had the better of the first half and might have had two penalties but it was a different story later as Bournemouth controlled the game.
Stephen Purches had an early shot blocked by home goalkeeper Ben Foster, and at the other end Neil Moss was equally on form as he saved a header from Chris Llewellyn and turned a shot from top scorer Juan Ugarte one handed over the bar.
Ugarte might have done better from his next opening set up by a neat chip from Mark Jones but he hooked wide from the edge of the box.
When Llewellyn and Ugarte combined, Moss dived bravely at the striker's feet to block his shot off the post.
Another left-foot drive from Andy Holt and a shot from Carlos Edwards went wide.
Matthew Mills was fortunate not to concede a penalty for a brisk challenge that sent Llewellyn sprawling and the Cherries escaped a penalty appeal again when Ugarte appeared to be pushed in the back when he was on a dangerous run inside the area.
Purches opened the scoring just before the hour when Edwards was caught out of position up field allowing Garreth O'Connor to cross for Purches to score.
The second goal came as a gift from Edwards, who was clearly tired after international duty in the Caribbean. He played a ball across the field into the path of Wade Elliott who had the simple task of chipping over the keeper.
Foster touched another shot from John Spicer round the post before substitute Chris Armstrong clawed a goal back.
The ex-Spurs striker might have gained a point from his next chance but fluffed his shot six yards out.