They use a kind of post-process antialiasing, unclear which one specifically but probably FXAA, and they use MSAA. Since DAI uses the Frostbite engine and its deferred lighting system, MSAA is pretty performance heavy. If you can tolerate barely perceptible blur, MSAA is the first setting you should nudge down in order to get extra frames in DAI.
What resolution and settings are you trying to play on? Dragon Age Inquisition was released on PS4/Xbox One, so its highest graphics settings are going to at least make use of the hardware in those consoles. From a pure hardware perspective, the PS4 is close to if not a bit higher than the Radeon 7850/265 (a bit hard to match, since the PS4 has more stream processors but a lower clock speed) and the Xbox One virtually the same as the Radeon R7-260, with a somewhat lower clock speed. All of which are better than the 7770 (which has the same chip as the 250). Twice the geometry power, more texture units, more stream processors, and in the case of the PS4, much higher memory bandwidth as it uses a 256 bit memory bus and GDDR5 memory.
If you don't think it's worth it to spend big bucks on the premium, high-end graphics cards, that's understandable. But increasingly, games are going to be tailored to the capabilities of the PS4 and Xbox One. If you want to keep parity with that, I would recommend upgrading to something from the R7-260 line (for parity with the Xbox One) or the R9-270 line (for parity with the PS4).
I haven't had the chance to actually play the game yet, but from what I've seen, the performance requirements are worth it. The art and detail is much better than Dragon Age 2.