The Heineken Cup quarter-finalists are currently nine points behind fourth-placed Sale Sharks with just six league matches left.
Successive defeats against Gloucester and Wasps have given Irish a lot to do, starting with Saturday's tough examination at Bath.
Smith said: "The outcome of our last two matches has been disappointing and has been a setback for our hopes of a top-four finish.
"We believe we still have a chance of making the play-offs. We are going to have a crack at it.
"That means winning five out of six games, which we are capable of."
After Bath, who will move top above Gloucester if they win this weekend, Irish face Premiership appointments with Bristol, Newcastle, Harlequins, Leeds Carnegie and Sale.
"Bath are among the in-form teams in the league at present and are playing good rugby," Smith added.
"Olly Barkey and Shaun Berne are good players that are managing games in a clever way behind a strong pack that provides tough opposition, especially at the Recreation Ground.
"We will have to produce an outstanding performance to stop them running the show if we are to have a chance of coming away with any points."
Irish though, are boosted by the return from injury of former Bath and England centre Mike Catt - he partners Seilala Mapusua in midfield - and there is also an opportunity for promising number eight Richard Thorpe, who replaces Juan Leguizamon.
Bath have four England internationals back in their pack, with hooker Lee Mears, prop Matt Stevens, lock Steve Borthwick and flanker Michael Lipman all returning from RBS 6 Nations duty.