Yes, it is actually a very good CPU for games. It can either match the capabilities or beat higher priced Phenom II X4 CPU in games (prior to the price cuts when the FX was released). It can also beat FX CPUs since the FX series seems to be marginally slower than the Phenom II X4 in some games; which is why the FX is considered a "side-grade" rather than an upgrade from a Phenom II X4.
The i3-2120 lacks 2 things though....
1. - It is not a quad core CPU, but the vast majority of games only use 2 cores. But in very specific games (i.e. Battlefield 3) that can be a difference. I do not play BF3, but from what I've read if you do not have a quad core CPU, BF3 will render few characters on the screen.
2. - It cannot be overclocked, so if you have any aspirations of OC'ing your CPU, then you are out of luck with the i3-2120. Only the Core i5 & Core i7 "k" models can be overclocked like the i5-2500k and the i7-3770k. On the other hand most AMD CPUs are overclockable, and when OC'ed they can equal a Core i3-2120 in gaming performance, or exceed it.
If you are looking for a quad core alternative and a Core i5 is more than what you want to spend then consider a Phenom II X4 965 (probably around $115). At stock speed it gaming performance is below that of a core i3, but when OC'ed the performance should be about equal.
Also, the higher the resolution you play at, the less performance difference you will see in games because they are mostly GPU bound. So while playing game "x" with GPU "y" at resolution "z", the FPS difference between a Phenom II X4 965 OC'ed and a Core i3-2120 may only be 6FPS in favor of the Core i3.