Olivier Giroud has been the Gunners' only recognised centre-forward this season, with Arsene Wenger using Nicklas Bendtner as back-up and Lukas Podolski currently out injured.
And as fears grow over the France international's workload, Gazidis feels they need to make a new striker a priority during the winter window.
"We will see where Arsenal will be in January and whether or not the club will invest in another forward," he told L'Equipe newspaper.
"Olivier Giroud won't be able to go on all alone until the end of the season. We need to buy."
Meanwhile, full-back Bacary Sagna believes Arsenal have now instilled a fighting spirit in the side and are not in fear of anyone as they aim to claim their first trophy since 2005.
"Technically we had a very strong group of players, which is now even stronger with Mezut Ozil. But we were missing the physical impact in the Premier League. We were missing the fighting spirit," he said.
"Maybe we thought our technical ability would be enough to make the difference. This time, we don't fear anyone."