Smith was furious with the decision to award Bernardo Silva's 79th-minute breakthrough goal, believing that Rodri should have been flagged offside in the build-up.
Rodri was stood in an offside position initially but moved back onside to dispossess Tyrone Mings and set up Bernardo's effort from range.
Villa's players protested to referee Jonathan Moss and Smith remonstrated with Coote on the touchline as both sets of players prepared to restart play.
Moss showed Smith a yellow card for his comment to Coote, then showed a red when the Villa manager continued to protest the decision.
"I thought they’d at least go over to the screen as it looked a clear offside," Smith said in his post-match press conference. "When I relaised they weren’t I told the fourth official they had juggling balls for Christmas.
"You have an official who says offside then changes his mind. Hence the yellow card for saying about juggling balls.
"For the red, I said you should have interpreted the laws of the game earlier in the game. I felt I was unfairly treated. It looked like he couldn’t wait to give me a yellow and a red. I'm frustrated at myself."
Smith still did not feel that the officials had arrived at the correct decision after the final whistle.
"I still don’t understand rule," he said. "I don’t think anyone can tell me why it’s been given. What is Tyrone meant to do? Can you stand 10 yards offside now?
"Nobody in stadium feels that should have counted so you can’t tell me that should have counted."
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