Bogdan was at fault after he let a tame Ed Upson shot trickle through his grasp late in the game.
Bolton started the better side, although both teams struggled to keep possession in the early stages.
Chris Eagles thought that he had scored when he had the ball in the back of the net only to be flagged for offside.
The Trotters were given a glorious chance to take the lead when Medo Kamara played in substitute Jermaine Beckford, but the former Everton and Leeds United forward hit his shot straight at keeper Wayne Hennessey.
Instead, it was Yeovil who went in front after Upson's weak shot trickled past Bolton keeper Bogdan and, with 11 minutes to go, it seemed that the hosts were condemned to defeat.
However, Zat Knight squared the ball for Mills, who bundled home the equaliser despite protests from Yeovil players that the ball had gonr out of play before it reached the defender.
The result pulls Bolton off the bottom of the table thanks to Barnsley's defeat at Leicester City, but does little for either side's prospects of staying in the division.