Rashford, who does not turn 19 until October, has now scored on his debuts in the Premier League, Europa League and Manchester derby as well as England's senior side and now the Under 21s.
He fired England ahead from 18 yards on the half hour mark before he added a deflected second on 66 and completed his treble having earned and then converted a penalty six minutes later.
After omitting him from his first senior squad before the recent World Cup qualifiers, new England boss Sam Allardyce said Rashford's inclusion in Southgate's side "will be invaluable for us later down the line".
The former Sunderland and Bolton manager also suggested that he would not select players who were not playing regularly with their clubs.
Rashford has played only 19 minutes in the Premier League for us so far this season as he came off the bench to score the injury time winner at Hull last time out.
Southgate said before the game that he had been impressed with the teenager's mentality and would have been even more pleased by his performance.
"What we saw tonight was exactly what we've seen from Marcus all week, not just in the quality but in the way he's gone about work.
"He joined this group from the senior England squad with real humilty and maturity, you hope that's the case with a young player but it's never a given.
"So he's impressed all of us with the way he's fitted in" said the young Lions delighted coach.
In front of the watching Allardyce, Rashford did not take his demotion to heart and made a perfect start to his second string career with a powerful run and well-taken finish after half an hour.
He netted his second with a strike that took a slight deflection that put his side 4-0 up and made it five soon afterwards with a composed spot-kick following a handball offence.
Next year, he would be eligible to play at both the Under-20 World Cup in South Korea, where England have already sealed their place as one of the 24 countries, and the European Under-21 Championships in Poland.
"It was the first penalty I've scored since I turned professional, it's always a test of your nerve so it has to be my favourite goal so far.
"It was a good debut to have and I thought Norway made it hard for us despite the scoreline, it's always an honour to play for England at any level so I am very happy" said Rashford.
Also on target at Colchester were Chelsea youngsters Nathaniel Chalobah, Ruben Loftus-Cheek and Lewis Baker who rounded off the scoring to complete the rout.