Marcus Rashford repeated the trick from his Manchester United, Premier League and England bows by scoring on debut for England Under-21s.
Rashford was part of England's travelling party for Euro 2016 but missed out on Allardyce's maiden senior squad for Sunday's last-gasp 1-0 win in Slovakia, having failed to break into the United starting line-up under Jose Mourinho.
But, on his first Premier League appearance of the season, Rashford came off the bench to net a stoppage-time winner at Hull City as Mourinho's men made it three wins out of three in the top flight.
And the 18-year-old offered another demonstration of his goal-scoring instincts against Norway's youngsters at Colchester.
Streaming clear of a static backline, Rashford slotted home a composed finish to break the deadlock after 29 minutes, meaning he has now scored on his professional, Premier League, full England and Under-21 debuts.
Rashford netted twice in the second half to put Gareth Southgate's team 5-0 ahead and keep them on track to qualify for next year's European Championship.