Calvert-Lewin went close to a second before Lukaku looked to have responded perfectly to his critics when he found the net, only for a late offside flag to deny the former Chelsea man his 20th league goal of the season.
Asked if Lukaku will face a backlash from anxious Everton fans when the Toffees host Hull on Saturday, Koeman told English newspapers: "I don't think so because we need the support - every player who is starting needs the support".
"The fans love Rom".
Romelu Lukaku's agent was reportedly spotted leaving Manchester United's team hotel, where the club's executive vice chairman Ed Woodward was staying. "This lad might be able to play at the top", was Keith Briggs's assessment of the teenager's contribution then. That's one of his big qualities. "Of course we like to keep our best players and we will do the maximum to keep these players but the final decision will be with the player himself". "We're in a good position in this project and you don't sell your best players".
One of them, 20-year-old Dominic Calvert-Lewin opened the scoring after eight minutes on Saturday with his first Premier League goal - turning in a cross from teenager Tom Davies.
Sam Clucas and Andrew Robertson both wasted chances for the visitors before Tom Huddlestone was shown a straight red card for a challenge on Idrissa Gueye.
"We kept the momentum".
And Koeman, who is in his first year at the Goodison helm, believes his team can improve upon this season's encouraging results. "We killed the game perfectly", the Dutchman said.
His double on Saturday, with both goals in added time, may not have been the decisive factor in putting the Toffees level on points with fifth-placed Arsenal, but it at least showed his professionalism can not be called into question. "It decided the game".