Wayne Rooney added a sweetly-struck effort late on to complete the rout, which sees Sir Alex Ferguson's men widen their advantage over neighbours Manchester City, who are not in action until Monday.
Kagawa's opener came when he tucked home Robin van Persie's flick-on in the box on the stroke of half-time, then Rooney played creator for his two tidy finishes after the break before adding one of his own in stoppage time - all in the space of 14 minutes.
It was an ideal way for United to warm up for their UEFA Champions League duel with Real Madrid in midweek and Ferguson rested Ryan Giggs for that clash as the Welshman was made to wait for his 1,000th senior career appearance, while Rafael and Rio Ferdinand were other notable absentees.
United's performance was low-key up to first-half stoppage time, with a brave Mark Bunn dive at Van Persie's feet and a low Michael Carrick effort that barely troubled the Norwich keeper the home side's only threat.
With Norwich's defence marshalled superbly by Bradley Johnson and Robert Snodgrass flying into challengers to expose any trepidation in United ranks, the visitors could feel well pleased with their efforts, but their good work was undone in those additional couple of minutes.
Antonio Valencia stepped neatly inside Anthony Pilkington on the edge of the area before floating over a rare left-footed cross. It was not exactly perfect for Van Persie, but when he stretched to control, the ball bounced down for Kagawa, whose shot caught Bunn by surprise as it crept in at the near post.
Despite Saturday's heroics, Kagawa could be one of the players to lose their place from the team that performed so well in the Bernabeu on Tuesday, while Patrice Evra is one of the certainties to be retained. It's little wonder, therefore, that Ferguson was so annoyed when Snodgrass launched into the French full-back in the dregs of the half. Evra was none too pleased either, although he did return for the second-half and Snodgrass collected a yellow card.
Jonny Howson had an effort deflected wide early in the second period before Grant Holt got above Evra and steered a cushioned header into Russell Martin's path, only for the full-back to lift his effort over from barely four yards.
Barely any pressure had been applied to the visitors' box when Rooney sped into the area 14 minutes from time off an impeccable Carrick ball and supplied the pass from which Kagawa doubled his tally for the season.
Kagawa was not finished though and another Rooney assist created a further opening for the Japan star, who again finished clinically to complete his hat-trick.
Rooney has a few of those of course. On this occasion, his goalscoring intervention was more brutal as he lashed a ferocious effort past Bunn in the final minute.