In truth a defeat would have been harsh on the Czechs who had 27 efforts on goal, forcing Mamardasvili into a fine double save as early as the third minute.
Bayer Leverkusen striker Schick may have done better with that early strike but another long throw from West Ham defender Vladimir Coufal soon brought about a chance, which Adam Hlozek bundled in at the second attempt.
However, referee Daniel Siebert was sent to his pitchside monitor and ruled the goal out for handball.
Another pitchside review gave Georgia their penalty for handball against Robin Hranac and Metz forward Mikautadze kept his cool to convert his second goal of the tournament.
The Czech Republic kept plugging away and got their rewards when substitute Lingr finally beat Mamardasvili, who made 11 saves in the match, with a firm header and Schick was poised to chest the ball over the line.
Ladislav Krejci wasted an identical chance moments later as he headed wide and Schick then limped off with an apparent calf injury which could yet prove decisive in the group’s conclusion next week.
Czech Republic’s 12 shots on target is the most by a team who failed to win a match at the Euros since records began in 1980.
They now face Turkey while Georgia – placed 74th in the Fifa rankings – take on Portugal knowing a win could see them through.
“When you are Georgia, you can’t pretend you can win the European Championship,” added Sagnol.
“Of course, if before the competition you would have told me you might play for qualification in the last match I would have signed immediately. We will play with ambition.”