Unfortunately, job titles vary from company to company, so a "programmer/analyst" at one place might just be a fancy term for "programmer" while at another place it might involve the creation of software specs and design docs, and in another it might involve budgeting and scheduling. So "Programmer analyst" is vague - you really need to list your skill set.
That said, can one transition from "programming" to "game programming"? Sure. Game programming can be thought of as a superset of programming: it's standard programming stuff plus the game-specific stuff (realtime input, graphics, networking, etc.). So, you've already got a foundation.
Keep in mind:
The game industry is insanely competitive. About half of the people in your class probably got into programming in the first place because of an interest in video games. Finding someone to pay you to make games is going to be an uphill battle. Especially if you have no field experience as a programmer, just a degree. (Real-world experience counts for a lot.)
There's no rule that says you can't be a programmer by day and make games on your own time at night, as a hobby.
In either case, take the time to learn how to do game programming (i.e. make some games on your own to teach yourself these skills). If you don't want to because it's too intimidating or just not fun for you, what do you think it'll be like when you have to do this stuff every day because it's your job?
If you are determined to find a game programming job, there are a few paths from here:
- Seek entry-level positions directly, right now. In the mean time, while you're waiting to hear back, bring yourself up to speed on game programming as best you can, and keep learning. If you are an outstanding, superstar programmer you might be able to pull this off; if not, realize you are competing against people who are, so it might take awhile.
- Seek an entry-level position at a non-game software company. Get some experience. When you feel ready to leave, start looking for game industry work.
- You can split the difference here: there are a fair number of "serious games" companies, gamification sites and similar companies that aren't "entertainment games" but are kinda sorta on the periphery and might be "close enough"... and the jobs there are usually less competitive.)
- Find a new school that offers a game-programming-specific degree, either as a graduate degree or a second bachelor's. This will probably be expensive and is certainly no guarantee, but if you learn best in a classroom environment this might be a way to push yourself in that direction. Some of these programs also offer part-time classes so that you can go there while also keeping a day job to support yourself. WARNING: there are some good schools out there and a lot of bad ones, and mistakes are very costly in time and money here, so make sure you're getting the value you want before you start if you go this route.