The forward scored twice in the first half, but was sent off in the second after tangling with Dundee skipper Stephen O'Donnell, who also got his marching orders.
O'Donnell appeared to fall on top of Russell, who pushed him away with his leg.
O'Donnell responded by shoving Russell aside, but the clash didn't look like a red-card offence by either player.
United boss Peter Houston said: "Johnny was fantastic and I thought he was very unlucky to be sent off.
"I have to see it again but I'm told he didn't do anything. He was told he was sent off because he'd lashed out but he didn't. We'll have a look at it again because we feel if he's not done anything then we have to appeal.
"If I see it and I don't think Johnny did anything then certainly we'll put in an appeal."
Dens Park gaffer Barry Smith saw a replay of the 72nd-minute incident before speaking and felt the decision was likely to be overturned without the need to appeal.
He said: "I don't think either should have been sent off. I think once the referee watches it, he'll realise that neither of them should have been sent off, and that will mean we don't have to appeal because he will rectify it."
Gavin Gunning got United's opener after 14 minutes, with Russell netting in 35 minutes and again three minutes later.