The Baggies thumped Thomas Tuchel’s side 5-2 last weekend with Robinson netting a brace thanks to two fantastic finishes.
But what should have been a dream day for the striker then turned sour when he was racially abused by some mindless Chelsea fans on Instagram.
High profile footballers are now being targeted by racists on a daily basis with Liverpool’s Trent Alexander-Arnold and Naby Keita as well as Brentford’s Ivan Toney also receiving abhorrent messages this week.
And Allardyce says both the government and social media companies are not doing enough to stop it.
“Callum received over 70 messages from 20 different sites (accounts) on Instagram,” Allardyce said. “We’ve had a great response from the police, our club is supporting him in every way we possibly can. This club has always stood up to discrimination.
“Within football, we’ve all moved forward in a very positive manner.
“But outside of the law is where these lads live, on Instagram, Facebook and Twitter.
“The solution can only be brought about by people in the hierarchy of the government and the people in the hierarchy of the businesses that run those sites.
“They are clearly neglecting their responsibility to help cut out this discrimination and racism.
“People hiding behind false sites and posting obscene messages, in this world we live in today, with the technology we have today, it is simply not good enough.”
Allardyce said Robinson was hurt by the messages but is strong enough to deal with them.
“He’s a strong character but it’s hurt him, of course, it has,” the boss continued. “The police have been brilliant interviewing him, they will do their best to look into it. The club are trying their best – we’ve written to Instagram, but I feel we’ll get a feeble response from them, not a positive one.
“Hopefully, his two goals are the most important thing and what he produced on the pitch and he can put this behind him.
“But certainly, it’s a responsibility higher up to put this right and to stop it.
“We as individuals or as a football club, the whole of football I don’t think can deal with this.
“I think it lies at a much higher level in government and across the world, I think it lies with the people who are running these sites are making trillions and trillions of pounds out of it.
“They have the technology to know what is happening and what’s going on so I think they have got the technology to stop it but they clearly aren’t doing enough at the moment.”
Back in January, a 49-year-old man was arrested on suspicion of racially abusing Albion’s Romaine Sawyers online.
Allardyce, though, feels it shouldn’t be the responsibility of the police to stop people posting vile messages.
“We did get one person arrested on a post that Romaine Sawyers had on his site – so that’s a positive,” the 66-year-old added. “The police have a department dedicated to it. But unfortunately for them, I would think they have limited resources to deal with this massive problem we are facing.
“They are obviously doing their very, very, best.
“But when you’ve got 70 messages from 20 sites, I would imagine that’s a much more difficult problem for that department of the police.”