The Sheffield Wednesday midfielder was shown a straight red for a tackle on Eric Lichaj.
Bannan, who was recalled to the Scotland squad earlier this week, was shown a straight red by referee Graham Scott following a late challenge on his former Aston Villa teammate Eric Lichaj, with the Owls already 2-0 up.
The 26-year-old has been one of Wednesday's key men this season and was instrumental in their win at the City Ground, providing a glorious assist for Aiden McGeady to open the scoring.
But the Owls face losing their summer signing from Crystal Palace for the next three matches following Scott's decision, which Bristol City boss Johnson and former Watford and Derby winger Smith both agreed was 'harsh'.
When asked if Bannan's tackle was worthy of a red card, Smith said: "No. It wasn't. It was a really tough decision. The only reason might be because his studs look like they're up there as they've gone in. If you watch closely it's his [Lichaj's] back leg he catches - it's minimal contact really. It looked very harsh."
And Johnson echoed his fellow pundit's view, adding: "I don't think it was a red card. It looks like it because he's raised his leg but the referee's got that one wrong."
Wednesday have now been issued four red cards in this season's Championship, though all but one of those have occurred in their past three matches.
Talisman Fernando Forestieri recently served a three-match ban for picking up two red cards in as many appearances, which despite also being deemed harsh were unappealable as they were both for two bookable offences.