Rovers dominated from the outset, with Lois Maynard coming close to a first-minute goal, as the visitors adopted a defensive approach.
There were few chances for either side until just after the hour when Luke Foster brought down Norwood.
He got up to take the kick himself, putting it to the left of keeper Max Crocombe and that proved decisive.
James Norwood's penalty earned Tranmere a sixth win in seven games as they overcame Southport at Prenton Park.