Zebo was on hand again 11 minutes later when Earls broke through the Dragons defence, moving the ball out wide before Simon popped it inside to Niall Ronan who crashed over the line for Munster's second try.
Dragons responded with two penalties kicked by winger Tom Prydie and outhalf Lewis Robling just before the half-time whistle.
Conor Murray replaced Peter Stringer after 50 minutes and made his presence felt straight away with a well worked try off the back of the scrum to extend Munster's lead to 13 points.
Three minutes later Zebo himself got on the scoresheet with Munster's fourth try, ROG kicking the convert to give Munster a 26-6 scoreline.
Dan Evans scored Dragons only try after 61 minutes with Robling kicking the points before Munster replacement Tommy O'Donnell sealed Munster's victory with the fifth and final try of the night.
Final score Munster 33 Newport Gwent Dragons 13.
Head Coach Rob Penney after the game "By no means were we perfect and they (Dragons) breached us a few times and made our lads work hard and they presented us with challenges so we certainly weren't perfect."
"Everyone is putting pressure on for a starting spot but it's really about the depth of players we have and it's critical for us to get through the season, guys can't play in every game."
"It's early days and we just have to keep our feet on the ground working on our game again this week."
Munster: I Keatley; D Howlett (capt), K Earls, J Downey, S Zebo; R O'Gara, P Stringer; M Horan, M Sherry, BJ Botha, Donncha O'Callaghan, B Holland; Dave O'Callaghan, N Ronan, P Butler. Replacements : D Varley, W du Preez, S Archer, D Foley, T O'Donnell, C Murray, D Barnes, JJ Hanrahan.
Newport Gwent Dragons: D Evans; W Harries, A Hughes, A Smith, T Prydie; L Robling, J Evans; P Price, S Jones (capt), D Way; I Nimmo, A Jones; D Lydiate, J Groves, T Brown. Replacements : S Parry, A Coundley, T Ryan, T Faletau, L Evans, L Davies, A Tuilagi, P Leach.
Referee: Marius Mitrea (FIR).