The 26-year-old moves to Ashton Gate on a free transfer after having his contract with Preston terminated by mutual consent in January.
Northern Ireland international Smith began his career in his homeland with Ballyclare, before spells with Sheffield United, Glentoran, Motherwell and Preston.
Smith will go straight into the squad for the trip to Brighton and Johnson is delighted to have finally got his man.
"I tried to sign Andy once before from Glentoran when I was at Yeovil, but Championship clubs came calling and he joined Preston," he told the club's official website.
"He's a hard-working striker, who is sharp with good movement. He can also play wide. We're delighted to have him on board."
Sporting Life