Hellberg has won both of his games since being appointed to replace Rob Edwards, who walked out on the chance of promotion to join Premier League strugglers Wolves last month.
Kick-off was delayed by 15 minutes as strong winds and heavy rain caused travel chaos that delayed Middlesbrough's arrival.
However, the visitors made a fast start once the action got underway as David Strelec fired into the roof of the net.
A three-goal blitz in 12 minutes before half-time then secured the three points.
Alex Gilbert marked his first appearance of the season with a thunderous finish before he teed up Morgan Whittaker to curl in his sixth goal of the campaign.
Riley McGree then took advantage of more slack Hull defending to prod home a fourth on 44 minutes.
Victory lifted second-placed Boro five points clear of Millwall in the battle for automatic promotion and cut the gap on leaders Coventry to seven points.
Joe Gelhardt reduced Hull's arrears from the penalty spot, but defeat meant that they missed the chance to climb into the play-off places.