Swedish striker Marcus Berg put fourth-placed Panathinaikos ahead after 32 minutes only to see Thomas De Vicenti level just after halftime.
The home team's quality began to shine through towards the end and Karelis struck twice in the final seven minutes to secure a third win in four games and give his side 20 points from 12 matches.
Panathinaikos are 14 points adrift of leaders Olympiakos Piraeus who won 4-1 at Panthrakikos 24 hours earlier. PAS are ninth on 17 points.
Following a stuttering start after a close season clear-out of players caused by financial problems, Giannis Anastasiou's young Panathinaikos team are starting to find their feet.
There was even more reason for optimism after president Giannis Alafouzos said they were set to benefit from new investment.
The owners of the club's basketball side, the Giannakopoulos family, have declared an intention to help the football team get back to its best.
"We are very happy with this development as they are a robust family who have already invested heavily in the basketball division to the tune of around 100 million euros ($135.24 million)," Alafouzos told reporters.