Second-half goals from Gary Teale, Neil Mellor and Clinton Morrison secured a 3-0 win that put the skids under Championship side City and gave the League One club a home tie with either Wycombe or Hereford in the fourth round.
Irvine was delighted with the performances of debutant defenders Michael Morrison and Reda Johnson, who helped soak up a lot of first-half pressure.
But once Teale broke the deadlock on 48 minutes Wednesday took control to leave Irvine with a broad smile of satisfaction.
"I've just been told that it's 10 years since we have got this far in the competition and that was a huge surprise to me," he said.
"You would think a club of this size and with what it has achieved in the past would have done better, but it makes this result even more pleasing.
"I told the players at half-time that they were better than they were showing. We gave the ball away too much and did not show enough self belief.
"The second-half performance was tremendous. Goals change games and by the end everyone wanted the ball and every player was showing composure on it.
"I'm not bothered who we get in the next round, but we have had two away ties out of three so far and it would be nice to be drawn at home.
"Michael Morrison and Reda Johnson could hardly have hoped for better debuts. We won't get carried away by one performance but defensively we looked very strong."
Chris Sedgwick supplied the right wing cross for Teale to head the opening goal and Mellor headed the second from a Daniel Jones centre seven minutes later.
Morrison ended any doubt about the outcome by shooting a close-range third from Mellor's cross on 65 minutes and Wednesday could have had more goals as City folded alarmingly.
Home boss Keith Millen admitted: "We were weak mentally after falling behind and the dressing room was too quiet afterwards.
"We looked a bad side in the second half and it showed we still have a lot of work to do over the rest of the season. It was a damaging result financially for the club and just how bad we will only know when we see who Wednesday draw in the next round. I have sent all my players home with a DVD of the match and will be looking for a response from them at our next training session."
Bristol City: James, Skuse, Caulker (Fontaine 9), Cisse, Nyatanga, Adomah, Johnson (Ribeiro 63), Elliott, Campbell-Ryce, Pitman, Stead, Ribeiro (Clarkson 73). Unused substitutes: Gerken, Stewart, Haynes, Sproule.
Sheffield Wednesday: Weaver, R Johnson, Purse, M Morrison, Jones, Sedgwick, Miller, O'Connor, Teale, Clinton Morrison, Mellor. Unused substitutes: Jameson, Spurr, Heffernan, Potter, Beevers, J Johnson, Palmer.
Referee: D Sheldrake (Surrey).
Man of the match: Neil Mellor.
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