The comprehensive win takes United, temporarily at least, 15 points clear of Manchester City and it is growing increasingly difficult to see where United are going to slip up to give their rivals an opening.
United had to wait until first-half stoppage time to make the breakthrough as Kagawa grabbed his first goal, before a late salvo saw the Japan international grab two more and Wayne Rooney added a fourth, all in the final 15 minutes.
Despite the crucial second-leg of their Champions League last-16 tie with Madrid to come on Tuesday, Manchester United manager Sir Alex Ferguson fielded a strong lineup. Perhaps most surprisingly of all was the presence of Robin van Perise, after the Dutchman went off with a hip injury last weekend.
But on this occasion, Van Persie took a back seat role as Kagawa grabbed the spotlight.
Van Persie did play a role in the opening goal as he tried to take down an Antonio Valencia cross from the right but the ball fell to Kagawa and the Japan international volleyed home at the near post.
The home side were again forced to wait to add to their lead, though as Norwich sought to build on their first win in 10 Premier League games against Everton last time out.
But United proved too good as their quality shone through late on. Kagawa added a second with a clever finish after Rooney beat a defender down the right of the area and pulled the ball back to the penalty spot.
And Kagawa completed his first hat-trick since his move from Borussia Dortmund with the pick of his goals. Danny Welbeck laid the ball into Rooney’s path on the edge of the box and the England forward found Kagawa with a lovely first-time ball that the attacking midfielder took into the area before dinking a fine finish over the on-rushing Mark Bunn.
Having contributed two assists, Rooney deservedly got himself on the scoresheet with a wonderful strike. From more than 25 yards, Rooney drifted past a challenge before looking up and unleashing a pinpoint strike that beat Bunn into the top corner.