Short answer: Probably.
Medium answer: Depends on the game.
Long answer: I'm pretty sure it'll do fine. Driving a 1080p monitor with lots of HLSL going on probably won't work at full speed.
I have a 3850 (I think) and it is fine on my 1366x768 monitor for all games including HLSL stuff. On my 1080p TV, things do not go so well with HLSL enabled at all. Things run fine without HLSL, though.