Robert Snodgrass regained the lead for the hosts with a flawless free kick twenty minutes into the second period as euphoria erupted at the KCOM Stadium.
"I hope that goal gives Ross a boost", Koeman, whose side remain seventh, said.... We dominated the game, but sometimes you can't stop a player like Snodgrass.
"We knew the productivity of Hull from set pieces - eight of their 14 goals have come this way this season".
"We spoke about being more unsafe in front of the goal and to have better offensive movements and he was really important in the final part of attacking and scored an important goal for 2-2 in the second half".
Tottenham Hotspur are planning to approach for Everton midfielder Ross Barkley, who has got another 18 months left on his contract, in the January transfer window.
"It has been hard for them, but I don't for one minute think that they have dropped away from the challenge". I wasn't happy with our slow start, but we showed a really good attitude and reaction twice. We fought for everything. We had the better chances to win the game.
"Maybe you expect a win, and then it will be totally different".
Dawson pounced with a lovely strike for his third goal of the season although Everton would look askance at veterans Phil Jagielka and Gareth Barry. He was outjumped by Curtis Davies for Michael Dawson's opener and his best years are definitely behind him.
Everton pushed forward in search of an equalizer and almost found it in the 15th minute when Seamus Coleman sent a header against the base of the right-hand post.
Marshall's confidence was at least helped after the break when he tipped a strike from Romelu Lukaku - deflected by Dawson - on to the bar, as the second half began in lively fashion.
Goalkeeper Maarten Stekelenburg will miss the game as he continues his recovery from the leg injury he suffered in the Merseyside derby defeat by Liverpool on December 19, with midfielder James McCarthy also ruled out.
It was awarded after Leighton Baines had fouled the impressive Harry Maguire who had seized on an error by Idrissa Gueye and run deep into Everton territory.