Lukaku moved two clear of Tottenham's Harry Kane on 21 goals in the hunt for the Premier League's Golden Boot award with two stoppage-time strikes at Goodison Park, adding to earlier efforts from Dominic Calvert-Lewin and Enner Valencia.
Everton moved level on points with fifth-placed Arsenal thanks to the victory - having played two games more than Arsene Wenger's side - and Koeman was quick to praise Lukaku after a week in which the striker refused to sign a new contract with the Merseyside club.
"The fans love Rom. Everybody loves Rom," Koeman said of Lukaku, who has just over two years remaining on his current deal.
"He's the key player and the top scorer and he showed why [against Hull]. If he gets that space he will punish the opposition and that's one of his big qualities.
"Of course we want to keep the best players. We're in a good position in this project and you don't sell your best players."
Calvert-Lewin's maiden Premier League goal put the hosts ahead early on, though an injury to Morgan Schneiderlin resulted in a stuttering display from the Toffees.
However, a harsh decision by referee Paul Tierney to dismiss Tom Huddlestone for a lunge on Idrissa Gueye enabled Everton to wrap-up the points in style late on, although Koeman acknowledged that Hull presented a tough challenge.
"We kept the momentum. But the final result didn't reflect what happened on the pitch," Koeman added.
"The decision of the referee about the red card was too harsh I believe. But it gave us more space to kill the game and we did that.
Hull boss Marco Silva, meanwhile, agreed with his counterpart's assessment of the red card incident, insisting that his side did not deserve to lose by such a margin.
"We didn't deserve this result and we didn't deserve what happened," Silva said. "I don't want to talk about [the red card] too much. We didn't deserve this. It decided the game."