Where would you go? We're celebrating our third wedding anniversary (and having some alone time away from our kids) on a short jaunt to MSP. We're staying near 2800 Chicago Ave S. Have car and will travel. We love French, Italian, Middle Eastern, just about anything except for Indian or Mexican. We'd enjoy a superb glass of wine, too.
I second all TDQ's suggestions (although Bar La Grassa is a little more "dinner with friends" than "anniversary" to me). If the main dining room at La Belle Vie looks too expensive, they also have a lounge that is a gorgeous space (and has its own smaller, less expensive tasting menu). Meritage is definitely further, but probably actually would not take much longer to get there at dinner time since most of the drive would be on the freeway. I love the wine list at Meritage.
Since you say you like Middle Eastern, I would recommend you add Saffron to your list of possibilities. I have been very impressed with the food the last few times I have been there. One other idea would be Al Vento in South Minneapolis. It is a cute little space and I think their pastas are very good.
Happy anniversary! Sounds like you'll be staying near Midtown Global Market, which I don't think has any great anniversary dinner options (more of a casual lunch place), but do pop in to Salty Tart Bakery at Midtown Global Market for coffee and breakfast pastries or a grab and go sandwich for the road. The American Swedish Institute on Park Avenue just a few blocks from there also has a lovely cafe, Fika, which is nice for breakfast pastries to go.
As far as your dinner, you will need your car to go to dinner anywhere of anniversary caliber.
If you want super romantic and quiet, La Belle Vie. Or if you don't mind a little cavernous and loud, Bar La Grassa (it isn't a bar, but it has one). Both in Minneapolis.
I love Meritage in downtown St. Paul for dinner, but it's a little bit of a drive from where you are.
These are all places that fit your French/Italian vibe that we've celebrated our anniversary, which means they aren't the latest and greatest *it* places, though they were not not so long ago, and they are still solid.