Researching an article that pours complimentary drinks as a matter of policy. For instance, here in Colorado, Boulder's Frasca follows the Friulian tradition of the tajut. A fellow 'hound also told me about a place in Phoenix that serves gratis glasses of Sambuca after the meal.
I assume most places that do this will be places that emphasize prix-fixe tasting menus and such. In short, the price of the drink, be it an aperitif or a sparkler or a digestif, is technically probably covered by the price of the meal. Even so, if you know of places that do this anywhere in the country, I'd be much obliged to hear about it.