Kee fired a hat-trick inside the first 18 minutes and went on to score five times in the first half.
Chris Zebroski and substitutes Ashley Yeoman and Danny Stevens joined in the fun, adding well-taken goals.
And it was only a series of saves by Tivvy keeper Chris Wright which stopped Torquay running up a double-figure score.
It was clear from the start that United's attitude matched their extra quality, as they challenged for every 50-50 and gave Tivvy no time on the ball.
They had to wait only seven minutes for a breakthrough.
Kee chested down a loose ball on the left, beat Adam Faux on the turn, cut inside and fired a right-foot shot inside the far-post from 20 yards.
Tiverton had a brief respite from the siege in the 15th minute, when Russell Jee floated a long free-kick to the far post, where Faux headed wide.
But normal service was resumed two minutes later. Kee chased a long ball from Mark Ellis, beat Wright's charge off his line and, after being pushed wide to the right, finished from a narrow angle for 2-0.
Kee completed a quick-fire hat-trick in the 18th minute. Zebroski crossed from the right, Kee controlled the ball, held off two defenders and fired a 15-yard shot past Wright's outstretched left hand.
In the 24th minute Kee turned provider, setting up Zebroski for number four.
Tiverton were forced to feed off scraps, but David Steele did well to work room for a cross in the 26th minute and Joe Bushin glanced a header wide from eight yards out.
But Kee was soon back on the scoresheet. First he brought a fine save from Wright with a 22-yard free-kick and then he reacted quickest to a loose ball and scored with a sweet overhead kick.
Ellis came up with a couple of good pieces of defending, especially to block a Bushin header, before Kee struck again. This time it was an unstoppable opportunist volley to make it 6-0.
Tiverton made four changes at the break in a bid to stem the tide, with ex-Torquay man Dean Stevens coming on for Adam Mortimer. Dean Coppard, Colin Marshall and Tom Gardner also joined the action.
But Yeoman, who also came on at half-time, added the seventh in the 51st minute.
Kee, still thirsty for goals, tested Wright with a rasping left-footer and then headed another sharp chance over.
Yeoman also had Wright scrambling back to his line to smother a goalbound effort.
Kee's strength saw him pick himself up from a tackle, wriggle free and then shoot straight at Wright when he seemed certain to grab his sixth goal after a good break from midfield by Danny Stevens.
Gulls keeper Potter was at last called into action in the 65th minute, when Marshall slipped Dean Stevens away down the left for Tiverton and the stopper did well to tip away the winger's strong cross-shot.
Town sent on veteran midfielder Mark Saunders for Bushin midway through the second half.
But United continued to created chances at will and they scored an excellent goal to wrap up the scoring in the 72nd minute.
Danny Stevens started the move with a beautifully-weighted through ball to Kee on the left. Kee then picked his moment as Stevens arrived on the edge of the area, delivered the pass and Stevens controlled before volleying past Wright.
It took a good save off his line by Wright to stop Kee scoring his sixth goal and the Town goalie also dived to his left to smother a long-range effort by Eunan O'Kane.
Despite the scoreline, it was little surprise when Wright was announced as Tiverton's man of the match.