The forward netted two spectacular goals as the Baggies ran out convincing 5-2 winners at Stamford Bridge on Saturday.
In honour of his newborn son, Robinson celebrated those strikes with a celebration where he pretended to rock a baby.
But while the 26-year-old is popular with his team mates and Baggie's fans, he has struggled for game time since the turn of the year.
Prior to the Chelsea match, Robinson had been an unused substitute in each of Albion's last seven outings.
And Allardyce was delighted to see the forward seize his chance when introduced from the bench against Thomas Tuchel's side.
“That's what he (Robinson) was brought in for, to score goals," the boss said.
"And the quality of his goals showed he does have the ability to produce that type of finishing in the Premier League.
"I’m absolutely delighted for him to be challenging the rest of the players for a place in the side.
"You want your substitutes to make an impact and what a massive impact he made.
"Let's hope he can keep it up because we need finishers to convert the chances we are creating if we are to stand any chance of getting even closer to safety.
"It has to carry on. It can't stop. It has to carry on."
While Robinson hasn't had many minutes for Albion recently, he did play twice for the Republic of Ireland during the international break.
And Allardyce felt those games probably helped him ahead of the Chelsea match.
"It was probably a good thing for him that he went away and played a couple of games for his international team," the boss added.
“He had been a bit short of games because he hadn’t been involved as much as he would have liked.
“But what we saw at Chelsea was what we haven’t been since we’ve been here.
“That’s the quality of his finishing. “If he can maintain the quality of finishing he maintains a place in the team, there’s no doubt about that."