The Blues moved six points clear of the drop zone after Wade Elliott's first-half goal sank Bristol City 1-0 and condemned the Robins to relegation.
Clark said: "We are not quite mathematically safe and it would be nice to take care of that with a good result in front of our own fans on Saturday.
"We owe them a big performance because we have not been at our best in recent home games."
Elliott swept home a right-footed shot after 16 minutes after Nathan Redmond's mazy run had unlocked City's defence.
But Birmingham had to withstand heavy second-half pressure from a spirited home team.
"My two centre-backs were fantastic," added Clark. "We had to defend well because we didn't pass the ball as we can.
"Our good away form has been the result of being solid as a unit and that was evident again in this match.
"Now I want a really big effort at St Andrew's because there are a lot of positives to take from the second half of our season."