Victory over promotion-chasing Bristol City on Easter Monday moved the Owls up to ninth with two matches left to play.
The result kept alive Wednesday's faint hopes of finishing in the play-offs. They trail sixth-placed Derby County by four points.
But Dawson, who has deputised admirably for the injured Keiren Westwood in the last two outings, believes Wednesday have left themselves with too much to do in the battle for promotion.
Nonetheless, the academy graduate has spoken of the importance of the Owls ending a turbulent campaign on a high.
Dawson told The Star: "I know it (the play-offs) is not mathematically over but it will be a big ask now.
"We will keep giving it everything.
"With the manager and staff, we are driven every game to get three points."
After Saturday's trip to Preston North End, the Owls sign off their season with a Hillsborough tussle with Queens Park Rangers on Sunday, May 5.
"We want to finish with two wins and see where it takes us," stressed Dawson.
"If we don't get into the play-offs, which looks unlikely, we can still try and carry some momentum into next year and look to hit the ground running.
"It is about finishing as high as possible."
Bruce has transformed Wednesday's fortunes, overseeing just two defeats from his 16 matches at the helm.
Dawson said: "The lads have been on a great run.
"The manager demands that we work hard every day in training.
"We work differently to what we have done in the past. It is a lot more physical in training.
"The mood is good and we are going into every game trying to win."