Irvine took over at The Hawthorns in June on a rolling one-year contract, following Pepe Mel's departure, but his short tenure came to an end on Monday.
Assistant Rob Kelly will take the reins for West Brom's trip to West Ham on Thursday, with the club set to appoint a permanent successor later in the week.
West Brom technical director Terry Burton said the decision to relive Irvine of his duties was "taken with regret but sadly driven by necessity".
"We appointed Alan in the summer convinced that we had taken on one of the foremost coaches in the UK and nothing that has happened since then has altered our view," Burton said in a statement.
"The individual progress of our players such as Craig Dawson and Saido Berahino are testament to that.
"But sadly that simply has not translated into results and they remain the ultimate currency of Alan's position.
"Securing a sixth season in the Premier League is the over-riding target and sometimes unpleasant decisions have to be taken to serve that imperative.
"Alan has impressed everyone with his manner, dedication and diligence but he knows that results have simply not been good enough.
"We place on record our gratitude for his efforts and hold nothing but good wishes for his future endeavours.
"This was a decision taken with regret but sadly driven by necessity."
The former Sheffield Wednesday boss oversaw only four Premier League victories in the first half of the season, and West Brom fans called for the 56-year-old to be shown the door during Sunday's 2-0 defeat at Stoke City.
That was West Brom's third successive loss and proved to be the final straw, with the West Midlands club 16th in the table and only a point above the relegation zone.
