Manchester City's Sterling celebrated the visitors' fifth goal by cupping his ears towards the home fans on a night when another impressive performance to start Euro 2020 qualifying by the Three Lions was overshadowed.
"I just want to show them you need more than that to upset us and stop us," Sterling told Sky Sports.
"It's 2019 now, I keep saying it. It's a shame to see this going on and we can only bring awareness to the situation.
"It's now time for the people in charge to put a real stamp on it because you can fine someone, but what is that going to do?
"You've got to make it a bit harder, punish fans that can't come to games. Do something that's going to make them think twice. If their team can't play with fans, it's going to be difficult."
Sterling also tweeted a photo of his celebration with the message: "Best way to silence the haters (and yeah I mean racists)."
England manager Gareth Southgate said he heard racist chants aimed at Danny Rose and promised to report the issue to UEFA.
"There is no doubt in my mind it happened. We will make sure that is reported to UEFA because it's not acceptable," said Southgate. "We've got to make sure we support our players."
England among Euro favourites
On the field, the World Cup semi-finalists showed why they are among the favourites to be European champions next year.
Southgate's trust in youth was again rewarded as 18-year-old Callum Hudson-Odoi, who is yet to start a Premier League game for Chelsea, shone on his full international debut, while Declan Rice, 20, was also handed his first England start.
"Fabulous performance," said Southgate on Hudson-Odoi. "With young players it is always up to them. You can see the talent, his application has been excellent. He's got to continue that."
England are not used to falling behind these days, but showed a resilience to bounce back after Marko Vesovic turned inside Michael Keane and curled past Jordan Pickford on 17 minutes.
Keane made amends for his earlier error with a commanding header from Ross Barkley's free-kick on the half hour mark to settle England's nerves.
Nine minutes later, Barkley applied the finishing touch after Hudson-Odoi's positive run into the box and shot which his club teammate turned in from close range.
Further goals from Barkley, Harry Kane and Sterling rounded off an excellent start to qualifying for England after a 5-0 thrashing of Czech Republic on Friday.
Sterling scored a hat-trick against the Czechs at Wembley, but also played provider here as his cut-back broke perfectly for Barkley to coolly side-foot home.
The Manchester City winger then led a rapid counter-attack with only Kane for company and squared for the England captain to tap home his 22nd international goal.
And after scoring just twice in his first 45 caps, Sterling now has six goals in his last four England games as he raced onto Jordan Henderson's through ball before slotting low past the goalkeeper.