The German World Cup-winning forward previously hinted that he may leave in the January transfer window after becoming disillusioned at the lack of first-team opportunities at the Emirates.
However, he scored the winning goal in Wednesday's 2-1 victory against Anderlecht in the Champions League.
Mertesacker told The Telegraph: "[Podolski] is in a difficult period, but how vital he is, he showed with his clinical finish. You can wake him up any second in the day and he will score with his left foot. So that is a quality we need.
"We play a lot of football around the box, but when it comes to finishing and shooting, we lack that."
Podolski's goal against Anderlecht was his first in 185 days.