Playing in an empty stadium because of a UEFA ban, Dynamo's winner came courtesy of a good deal of fortune when Maccabi's Tal Ben Chaim deflected a free kick towards his own goal.
Goalkeeper Predrag Rajkovic could only push the ball into the path of Garmash, who accepted the 16th minute gift gratefully.
The goal was enough for Dynamo to leapfrog Porto, who lost 2-0 at Chelsea, and snatch second place in Group G.
It was a night of celebration for Kiev coach Serhiy Rebrov, who led the club to the last 16 for the first time in 16 years.
The 41-year-old former Kiev striker was part of the team that reached the second group stage in 1999-2000, a year after they made the semi-finals.
“It was a very tough game from the point of view of psychology as it is always hard to play without fans," Rebrov told reporters.
"It is a completely different story when you see the empty stands. We tried to hide our anxiety but the stakes of the game were high and the lads were constrained in their actions.
“We should take our time to enjoy the moment as we have not been in the knockout stage for 16 years. Half of our team had no Champions League experience and I think it is a great success.”
UEFA had ordered Kiev to play two home games in an empty stadium after racist attacks on Chelsea fans at an earlier Champions League match.
They will also have to hold the home leg of their last 16 tie behind closed doors.
(Writing by Ian Chadband; Editing by Rex Gowar/Toby Davis)
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