The Tigers were pegged back twice versus Ronald Koeman's men at the KC Stadium on Friday evening.
Speaking to SkySports after the game, The Tigers skipper bemoaned a missed opportunity to pick up a valuable three points, after dominating large portions during the 90 minutes in Humberside.
Hull City's Michael Dawson celebrates scoring their first goal
After Dawson had given the hosts the lead against The Toffees, a goalkeeping howler from David Marshall saw Ronald Koeman's men go into the break level.
Robert Snodgrass scored his sixth of the campaign, with a delicious free-kick from 22 yards out to regain the lead, before Ross Barkley's 84th minute header saw Everton take a point back to Merseyside.
The result leaves Hull still mired in the bottom three - despite climbing a place up the table. The Hull skipper reflects ruefully on the result and could yet be decisive come May.
Dawson told SkySports: "Obviously you go ahead twice at home - to draw the game it's a big disappointment to concede the late goal. It takes us off the bottom, we'll try and build on that."
"We played well on the first half to take the lead so early on and to concede just before half time it knocked us. We got into half time and got under a little bit of pressure, but Snoddy got the goal. We're just disappointed we couldn't hold on for the three points."
Hull City's Robert Snodgrass scores their second goal from free kick
Hull now face a tricky trip to in-form West Brom on Monday, before a home FA Cup Third Round tie with fellow strugglers Swansea City next weekend.
On the subject of a impending relegation battle on the horizon, Dawson added, "Results haven't quite gone our way in recent weeks, but the performances are there and we've gotta keep believing."
"The manager said that and the groups of players - you've seen it tonight - there's one thing; We'll keep fighting and we'll keep believing right till the end."