The visitors were awarded a 69th-minute spot-kick when Johnny Hunt fouled Adam Mekki in the area, allowing Norwood to confidently despatch from 12 yards.
Rovers, who move up to fourth in the National League, were good value for their win, being rarely troubled.
Tranmere's players thought they should have been awarded a second penalty in injury time when Steven Jennings' shot struck the arm of Ben Heneghan.
Chester, who started the game playing three at the back, were disrupted by the early blow of losing captain Ian Sharps, playing against his first club, with a hamstring injury.
Steve Burr's side stay 15th, but are now only six points clear of trouble, going into successive home games against the two sides immediately above them, Bromley and Aldershot.
Chester manager Steve Burr told BBC Radio Merseyside:
"We were dealt a terrible blow after just 10 minutes when we lost Ian Sharps because I then had to rearrange the whole team.
"We stuck Sam Hughes at right-back because I thought playing 4-4-2 would be better, but I never want to see those situations happen.
"Ian was probably the worst player we could have lost because of having to rearrange. But I thought Sam did really well. He's only young and still learning the game."
James Norwood's second-half penalty earned Tranmere a derby win at Chester.