Christian Leali'ifano's kicking issues linger, with the sideline conversion cruelly taking the top of the left-hand upright after an earlier missed conversion.
Mogg slotted a late penalty from 55 metres to seal the win, having earlier pushed two long-range kicks wide, including one from 62 metres in the eleventh minute.
Hard-working Lions backrower Jaco Kriel was rewarded with a second half try, while fullback Andries Coetzee scored a late consolation try.
The Brumbies had led 22-6 at halftime, courtesy of a dominant twenty-minute period where they piled on three tries.
Lions first five-eighths Elton Jantjies opened the scoring and winger Ryan Combrinck added a long-range penalty from 59m out to put the home side up 6-0 after 13 minutes.
Leali'ifano opened the Brumbies' account with a 15th minute penalty.
Centre Tevita Kuridrani scored the first try of the match in the 18th minute after several phases of sustained pressure and Ita Vaea crashed over only minutes later, after a Mogg counter-attack put Henry Speight into space down the right flank.
Leali'ifano then left the Lions shell-shocked, intercepting a pass on the Lions' 22m line and scoring under the posts to push the Brumbies' lead out to 22-6 after 27 minutes.
Forced to abandon their territory game in the first half, the Lions regularly made ground running out of their own territory, only for handling errors to continually let them down.