Soon after Arouna Kone's second minute opener for the visitors, referee Lee Mason was seen consulting with managers Roberto Martinez and Keith Curle, and a message then came over the public address system announcing that any racist abuse would not be tolerated.
Toffees boss Martinez later said it was his understanding some "racial remarks" had come from the Warwick Road End stand as Kone and Steven Pienaar celebrated the goal.
Opposite number Curle said: "It is not what we want inside Brunton Park, inside football, or inside society. It is not accepted."
A statement from Cumbria Police issued after the game said: "Cumbria Constabulary are investigating allegations of racist behaviour?directed towards Everton players following the first goal.
"Hate crime will not be tolerated in this county and any allegation will be fully investigated.
"Officers are making enquiries and the force is asking for those with information to get in touch."
An FA statement on Sunday evening said: "We have received the referee's report regarding the alleged incident during today's game between Carlisle United and Everton.
"We have contacted both clubs this evening and will be conducting an immediate and thorough investigation.
"The FA strongly condemns any discriminatory behaviour and will continue to work with partners across the game to combat it whenever it arises."
Both clubs have pledged to work closely with the authorities in their investigations, with Curle saying: "To any supporters who were next to people and heard people - please step forward and identify them."
Martinez praised Mason for his handling of the matter, and added: "They (Kone and Pienaar) are okay.
"I think they were sensitive and hurt at that moment - probably they were fearful about the game developing into a difficult afternoon.
"But it was a one-off incident.
"There is no room for that sort of behaviour, but I must stress it was a very small minority and I want to believe that it was just an emotional reaction to the disappointment of conceding a goal.''
Everton went 2-0 up in the 14th minute through Aaron Lennon, and Ross Barkley's deflected strike in the 65th subsequently sealed the victory that has set up a fifth round trip to Barclays Premier League rivals Bournemouth.
A bumper crowd turned out for what was only Carlisle's second game back at their own stadium since it was caught up in devastating floods.
And Curle, whose Sky Bet League Two side were the lowest-ranked outfit left in the cup, said: "A lot of credit has to go to a lot of people who worked to get this fixture on.
"And to have 17,000 people here was phenomenal.
"I think it was only 57 or 58 days ago that the place was decimated because of the disaster of the flood, so to be hosting football - and hosting Everton - here was excellent.'
Source: PA