From a girl who's held a few joysticks in her day
1. Make the night about her-
Order in some food, cuddle up on the couch, hell, light a few candles. For this to work, she's going to need to think this night revolves around her pleasure. The key to success lies in how you introduce the idea. Say the wrong thing and you'll be shutting down before you even hit the power button. Try to speak her language. I suggest something along the lines of:
"I want to share my interests with you so that we can understand each other better."
"My buddies want me to hang out with them but I'm feeling a little low tonight. Let's stay in and eat junk food and snuggle!"
2. A little booze never hurts-
Of course, I'm not promoting non-consensual gaming. However, a little social lubricant will help to get even the most reluctant girlfriend over her inhibitions and into whatever you two want to play. Just make sure she remembers the whole thing in the morning, else your sorry ass is back to level 1.
3. Don't set your expectations too high-
Beware: the first time won't be perfect. It might even be a little painful for you. But if you make her happy on the first go, you can expect the experience to progressively improve. One day, you might even experience the same elation playing with her as you do with your bros.
4. Start slow-
In an ideal world, your girl would dive headfirst into this fantasy without any regret. However, you have to keep in mind that this is all new for her and it might be a little scary. You're going to have to warm her up to the situation to get her excited. Keep games like Katamari Damacy on hand for when the time is right. This bubbly, pretty game will keep her entertained while introducing her to basic game design and controller mechanics. From there you can move on to subtly more advanced systems.
An important warning- while cute videogames are a plus, avoid the Wii at all costs. She'll no doubt love Wii Sports or Mario Party 8, but the bastardized game-play won't improve her skills or change her attitude and will probably get you no closer to home base- all ur home base, that is.