Colin Nish was shown a straight red card for a challenge on Paul Dummett when the match was still goalless and a double from Steven Thompson - either side of a Ryan Conroy penalty for Dundee - and a first Saints goal for Dougie Imrie settled the issue.
Smith said: "I've not seen it again but anybody who knows Colin Nish knows he's not got a malicious tackle in him.
"He's a big gentle giant. I find it hard for it to be a sending-off knowing him, but the referee has got a decision to make and that's what he made.
"The pleasing thing for me is that we didn't just lie down and say we've got 10 men.
"It was a case of still having something to fight for and we fought for it. Unfortunately, we lost two goals near the end."