The 25-year-old, who scored 19 goals for the Prenton Park outfit last year, got Rovers off to a dream start after nine minutes, firing past Alan Julian from Connor Jennings' pinpoint cross.
Blair Turgott missed a gilt-edged chance to equalise for the hosts from six yards after 40 minutes. Tobi Sho-Silva seemingly did all the hard work, ghosting past Tranmere's backline, but the West Ham youth product hit his effort wide.
Four minutes later Norwood made it 2-0, prodding home Andy Cook's flick from Liam Ridehalgh's cross and that was enough for the Merseysiders to make their 500-mile round trip a fruitful one.
Report supplied by the Press Association.
Match ends, Bromley 0, Tranmere Rovers 2.
Second Half ends, Bromley 0, Tranmere Rovers 2.
Substitution, Tranmere Rovers. Jake Kirby replaces James Norwood.
Corner,  Tranmere Rovers.
Substitution, Bromley. Rob Swaine replaces Joe Howe.
Substitution, Tranmere Rovers. Adam Mekki replaces Andy Cook.
Substitution, Bromley. George Porter replaces Reece Prestedge.
Connor Jennings (Tranmere Rovers) is shown the yellow card for a bad foul.
Corner,  Bromley.
Michael Ihiekwe (Tranmere Rovers) is shown the yellow card for a bad foul.
Substitution, Bromley. Connor Dymond replaces Max Porter.
Corner,  Bromley.
Corner,  Tranmere Rovers.
Corner,  Bromley.
Corner,  Bromley.
Corner,  Bromley.
Second Half begins Bromley 0, Tranmere Rovers 2.
First Half ends, Bromley 0, Tranmere Rovers 2.
Goal!  Bromley 0, Tranmere Rovers 2. James Norwood (Tranmere Rovers).
Corner,  Bromley.
Dave Martin (Bromley) is shown the yellow card for a bad foul.
Reece Prestedge (Bromley) is shown the yellow card for a bad foul.
Corner,  Bromley.
Corner,  Bromley.
Corner,  Tranmere Rovers.
Goal!  Bromley 0, Tranmere Rovers 1. James Norwood (Tranmere Rovers).
Corner,  Tranmere Rovers.
Corner,  Bromley.
First Half begins.
Lineups are announced and players are warming up.
James Norwood's first-half brace was enough to give Tranmere a 2-0 win at Bromley to start their 2016-17 National League campaign in style.