The Gwent side scored five tries and made light work of the visiting French side with Wales centre Sweeney keeping the scoreboard ticking over.
With a tough cross-border clash against Bristol at the Memorial Stadium next Friday night this was just the sort of disciplined performance coach Paul Turner would have wanted.
Four yellow cards were shown by referee Rob Debney, but the aggression from both packs never spilled over.
In the four years of regional rugby, this was the first time a Welsh side had appeared in the European Challenge Cup.
It might be the second-tier competition, but it was still graced with the presence of Wales coach Gareth Jenkins and team manager Alan Phillips, no doubt checking on the form of several Dragons ahead of Wednesday's announcement of the squad to face Australia.
Bayonne brought ex-French international centre Richard Dourthe and made a lively start, Kevin Morgan had to scramble back to cover a deep kick to the corner.
The Dragons were rewarded for some early pressure with a penalty which Ceri Sweeney had no trouble landing from 35 yards out making it 3-0 with 10 minutes gone.
And their pack soon marched the French back over their own line with Wales No 8 Michael Owen touching down after 18 minutes as Sweeney landed the conversion for a 10-0 lead.
Five minutes later and Sweeney stepped up again to slot home another penalty to push the Dragons out to a 13-0 lead with a quarter of the game gone.
A hand in the ruck by Bayonne in the shadow of their posts had the same conclusion with Sweeney landing the kick as the Dragons dominated the game at 16-0 with 26 minutes played.
Just to remind the Rodney Parade crowd they were there, Bayonne No 10 Jon Iturriria dropped a towering goal from 45 yards.
Bayonne had suddenly come to life and started taking the game to the Dragons.
Wales lock Ian Gough was shown a yellow card for over-vigourous use of the boot as the Welsh side found themselves down to 14 men with half-time approaching.
Bayonne joined them at 14 men when lock Cedric Bergez aimed a kick at his opposite number Luke Charteris' head.
In injury time, Sweeney smashed over his fourth penalty of the night from just over the halfway line.
Bayonne went down to 13 men when hooker Arnaud Heguy was yellow-carded for deliberate offside.
The French sent on a replacement front-row forward, but their scrum collapsed on their own line which led to a penalty try.
Sweeney banged over the conversion to make it 26-3 with 48 minutes gone.
Owen went to grab his second try of the night after picking up from the base of a scrum, but his offload went to Gareth Cooper who touched down with Sweeney landing the conversion.
Super sub Richard Fussell showed he had plenty of pace to round the defence and touch down under the posts for his side's fourth try which made the bonus point secure.
Fellow replacement Nic Fitisemanu added a fifth try with ten minutes left and Sweeney's conversion curled away from the posts and blotted his copy book.
Hercus poached an interception try in injury time from just inside his own half after a 60 metres sprint, but Sweeney missed the conversion again.
Dragons: K Morgan, N Brew, P Emerick, C Sweeney, A Brew (R Fussell 58 mins), M Hercus, G Cooper (Ireland 69 mins), A Black, B Daly (Crawford 69 mins), L Harrison (Corsi 70 mins), I Gough (Sidoli 69 mins), L Charteris, A Hall, C Charvis, M Owen.
Bayonne: C Fromont(Inigo 62), S Ternisien, R Dourthe, V Inigo, Y N'gog, J Iturriria, S Rouet, J-M Usandisaga (Iguiniz 40), A Heguy, A Loprestti (Garcia 40 mins), K Rudzki, C Bergez, J-J Marmouyet (J Galleco 44-54 mins), J Deen, C Ternisien (Baget 55 mins).
Referee: R Debney
Yellow Cards: Gough 34 mins, Bergez 39 mins, Heguy 44 mins, Dourthe 79 mins
Dragons: Owen, penalty, Cooper, Fussell, Fitisemanu, Hercus tries, Sweeney (4) con, Sweeney (4) pens
Bayonne: Iturriria drop goal