Bournemouth are up to 12th in the Premier League table while Brentford move into the top half on goal difference.
Story of the match
An attempted clearance from a Brentford corner fell directly to Mathias Jensen, but he hit his shot straight at the Cherries goalkeeper. He then saved from Damsgaard's drive two minutes later.
On 21 minutes, Marcus Tavernier gained possession on the right and eventually played in Jordan Zemura, who was felled by Ajer. Referee Thomas Bramall consulted with VAR and upheld his original decision.
Moments later, Baptiste played a sublime through ball to Toney, the Ivorian making a mess of his left-footed shot that allowed Neto to eventually gather.
Damsgaard then centered at the back post for Ajer, who rose highest, but scuffed his header over the bar. At the other end, Ben Mee got in a superb block to deny Tavernier.
Within five minutes of the restart, the hosts were nearly in front as Moore headed back across goal and Jefferson Lerma nodded on, but Toney tracked back to head off the line.
Frank made a triple change on the hour mark, introducing Dasilva, Yoane Wissa and Aaron Hickey.
Dasilva nearly made an immediate impact as he hooked a first-time pass to Toney, who got behind the Bournemouth defense, but his half-volley that flashed just wide of the post.
The same duo combined again only to see Mepham getting in a key block and the former Brentford defender made another close-range block on Toney and threw himself in front of Wissa's curler.
In the dying moments of stoppage time, the Cherries thought they had another penalty after a corner kick was headed down inside the area. Jensen tried to hook it clear, but the ball bounced off of his hand.
Bramall whistled for full time rather than award a penalty and both sides were left to settle for a point.
Player of the match: Chris Mepham
The former Brentford man played a vital role in helping Bournemouth keep the Bees off the scoreboard, winning first and second balls, made several crucial tackles and dealt with physical play brilliantly.