Results earlier in the day meant both sides had already booked their places in next month's European Under-21 Championship knock-out double-headers.
But the group - and in theory a more favourable draw as one of the seeded teams in Friday's draw - was up for grabs at Proact Stadium, and Stuart Pearce's side won it courtesy of Wickham's strike two minutes before half-time.
It was one of few chances England created in a lacklustre contest and in the end the Young Lions made hard work of the victory, with Norway having the better of the second half and unlucky not to equalise when Havard Nielsen's effort struck the underside of the crossbar and rebounded to safety.
Pearce made five changes from the side that started Thursday's 2-0 win over Azerbaijan.
Jason Steele returned in goal after missing out through illness in Baku, meaning Ben Amos replaced Declan Rudd on the bench.
Ryan Bennett came in at centre-half for the suspended Craig Dawson while midfielder Ben Marshall made his full debut in place of the injured Jason Lowe.
Wickham and Martyn Waghorn were preferred in attack ahead of Wilfred Zaha and Marvin Sordell, who were both among the substitutes.
Waghorn saw a thunderous left-foot shot pushed away by Norway goalkeeper Arild Ostbo in the sixth minute before Bennett headed over from the resulting corner.
Tom Ince looked like he would put England ahead when he sneaked in between the Norwegian defence and keeper on the half-hour mark. The Blackpool midfielder managed to clip the ball goalwards past Ostbo but it rolled off target.
Wickham showed Ince how it was done after 43 minutes.
The Sunderland striker latched onto a defensive mistake and, bearing down on goal, tucked away his chance well from close-range.
It was the 19-year-old's first goal since October last year.
Norway started to come into the game after half-time and substitute Yann-Erik de Lanlay shot narrowly wide on the hour and ten minutes later Nielsen rattled Steele's crossbar.
Norway almost caught England out again in the 73rd minute. Omar Elabdellaoui broke clear and flicked the ball wide to Mushaga Bakenga but his cross-shot was diverted over for a corner.
With nine minutes remaining, a strike at goal from Harmeet Singh 25-yards out had Steele stretching but the Middlesbrough keeper grabbed the ball at the second attempt.
Man of the match Danny Rose joined Shelvey in collecting a needless yellow card after kicking the ball away as England withstood some more late pressure to hold on for the win.