The SPL's bottom club went down 3-1 to Aberdeen at Dens Park last Saturday with what was their poorest performance since coming back into the SPL. It left the Tayside club nine points adrift at the foot of the table and Smith is asking his men for a reaction when they travel to McDiarmid Park in a bid to start the new year on a high.
"We know we let ourselves down. It was way below what we can do and that was the most disappointing thing about it," Smith said.
"So we have to be much better against St Johnstone than we were in the Aberdeen game. That goes without saying.
"That's everybody's aim and it's up to us to go out there and make sure that we are much improved."
Dundee know the second half of the season is going to be vital in their ongoing efforts to stay up following promotion after a seven-year exile.
The Dark Blues have suffered back-to-back home defeats including the Boxing Day loss at the hands of Celtic. But Smith insists they can still make up the ground on their nearest rivals to stay in the top-flight.
"It was always going to be difficult but our aim is still the same," he said.
"We're all in this together and the main thing is that we stay in the SPL. To do that we have to make sure we get enough points on the board which we can still do.
"One good thing is that we have a great spirit among the lads in the dressing-room which is important."