The Bees continued to look a threat going forward and Lasse Vibe burst through the home side's defence only to see his effort on goal blocked by the feet of Amos.
Amos was being kept busy in the Bolton goal, and the keeper was called into action once again to push away Swift's volley at the back post.
Brentford took their lead into the break and they almost doubled their advantage early in the second half when Jake Bidwell's header across goal dropped narrowly wide.
However, Bolton made their opponents pay for their missed chances in the 65th minute as Neil Danns picked out the bottom corner with a shot from the edge of the box to level the scores.
Within minutes the hosts had a chance to score another goal after the ball dropped kindly to Ameobi inside the box, but the striker could only fire straight at the keeper.
In the closing stages Brentford almost stole the points as Vibe saw an effort kept out by Amos, before Konstantin Kerschbaumer's shot was blocked, but Bolton managed to hold on for a share of the spoils.