Considine turned provider as the first half wore on, picking out Shaleum Logan inside the area for a headed chance which the defender failed to convert.
Aberdeen felt that they should have been awarded a third goal prior to half time when Adam Rooney's strike appeared to cross the line, although the officials disagreed and opted to wave play on.
A red card issued to Jesus Garcia Tena made Accies' task that little bit more difficult, and having pushed on to make the man advantage count Aberdeen once more had a goal ruled out after Consadine was adjudged to have fouled a defender in the build-up.
Incredibly, a third Dons goal was then flagged off when David Goodwillie converted only to find himself in an offside position, meaning that Hamilton still had a glimmer of hope in the final stages.
That was until McGinn superbly placed the ball beyond McGivern when played through one-on-one, coolly converting to round off the scoring two minutes from time at New Douglas Park.