Magpies defender Malick Thiaw scored the fastest goal of the season after just 52 seconds, and Lewis Miley and Nick Woltemade netted to put the result beyond doubt by half-time.
Thiaw added his second after the interva,l and Kiernan Dewsbury-Hall's strike for Everton was little consolation for the hosts.
At the final whistle, Howe and his players gleefully soaked up the acclaim of the travelling Newcastle fans after their first top-flight away win since April.
"That was really important as we haven't had that feeling enough or at all," Howe said.
"Very pleased today that we were able to put that right and give them that winning feeling to go back to Newcastle with.
"I'm just pleased I don't have to answer the dreaded questions on it.
"We have a lot of work to do to say we have solved some of the issues we have had, but it is a big moment for us. A step forward."
Asked whether a significant mental hurdle had been overcome, Howe added: "Time will tell. I'm not going to make any big statements, there is a long way to go."
Everton manager David Moyes refused to use the absence of key midfielder Idrissa Gueye, who was sent off for slapping team-mate Michael Keane in a rare win at Manchester United on Monday, as an excuse for the dismal display.
"It could be that, but it could have been the quality of Newcastle United. You mustn't underestimate them, they're a Champions League team," he said.
"They got off to a great start, we got off to a terrible start. They were much better than us, played better, and much more experienced than we are."