Before the Egyptian brilliantly cancelled out Hal Robson-Kanu's opener, Dean awarded Liverpool a free-kick after seemingly getting in Fabinho's way.
Usual protocol would see referee's award a drop ball in these circumstances, but Dean instead awarded the Reds a free-kick which eventually led to their goal.
The official rule states that if the ball touches the referee or another match official and goes into the goal, or results in a change of possession or a promising attack, a dropped ball will be awarded.
And speaking at half-time on Sky Sports' coverage of the game, Jamie Redknapp and Graeme Souness said that Baggies boss Allardyce would be seething with the officiating.
"It doesn't make sense why it's a free-kick to Liverpool," Redknapp said. " I don't quite get it and they get away with it because Conor Gallagher is out of position."
Souness added: "Sam will be fuming.
"I don't know what the referee is thinking. That should be a drop ball"
Salah's effort came after Robson-Kanu gave the hosts a shock lead as he raced onto Matheus Pereira's clipped ball over the top before tucking home with his left foot.
It was a a brilliant finish from the Welshman, who was handed his first league start of the season by Sam Allardyce at the Hawthorns.
Sadio Mane, who returned to Liverpool's XI after snubbing Jurgen Klopp's handshake following the victory over Manchester United last week, spurned a glorious chance to level proceedings little under 10 minutes later as his effort from just inside the area glanced wide.
But Salah equalised contentiously moments after as he went level with Harry Kane at the top of the Premier League's goal-scoring charts, with both players on 22 strikes.
West Brom were perhaps unfortunate not to have scored in the second half when Kyle Bartley nodded home, only for VAR to intervene and say Alisson's view was obstructed by a Baggies player.
But in what was one of the most memorable finishes in Premier League history, the Liverpool goalkeeper grabbed a last-minute winner for the Reds after coming up for a corner and glancing in from six yards to seal a 2-1 victory.