Knowing that they were in need of a win, the hosts came flying out of the blocks and had numerous early chances to open the scoring.
The Sky Blues were given a golden opportunity to make their pressure pay when they were awarded a penalty shortly after the break.
Taylor was adjudged to have fouled McDonald in the box but the Lions stopper made amends by saving Gary McSheffrey's spot kick.
To make matters worse for the struggling hosts, Millwall went ahead soon after with their first real chance of the match.
Taylor continued to act as Coventry's nemesis, again denying Clingan from a free kick as City's Championship status became even more insecure.
Andy Thorn's side are now five points away from safety with just two games of the season remaining.