Early tries from Burrell and Api Ratuniyarawa put Saints in control of the European Challenge Cup Round 3 encounter before James Fish scored on the stroke of half-time for a 19-0 lead.
Adam Warren hit back at the start of the second period for Dragons, but further tries from Burrell, Naiyaravoro (2) and Alex Mitchell wrapped up the bonus point home victory.
Back row Nic Cudd struck late on before Ollie Sleightholme made sure the English Premiership side had the final word.
Dragons made eight changes to the side that lost to Leinster in the league, with prop Lloyd Fairbrother and centre Jack Dixon reaching memorable milestones as they hit 100 appearances for the region.
It was the home side, though, that made a lightning quick start and, after Dragons knocked on from the kick-off, the ball was crashed up the middle before Burrell bundled over for the lead inside 80 seconds.
Dragons looked to respond with Josh Lewis taking a high clearing kick at pace before his neat chip in behind was taken by a galloping Dixon who burst over halfway.
At the other end, Sleightholme was only stopped up a superb double tackle from Richard Hibbard and Dafydd Howells as they shunted the home wing into touch.
Powerful running from powerhouse lock Ratuniyarawa secured try number two as he crashed through two tackles and stretched over the line after another incisive home surge.
Three quick-fire penalties saw Dragons enjoy a spell of pressure and repeatedly threaten the line before Saints turned it over and cleared mounting danger.
Dragons were dealt a double injury blow as the first half wore on as Hibbard (HIA) and Ollie Griffiths (knee) were forced off.
Adam Warren did brilliantly at the breakdown to win a penalty after Saints threatened from deep after two clearing kicks failed to find touch.
The hard-running centre then broke up the middle but his final pass to Amos was juggled and then fell to ground as the full back looked to run in.
Saints were reduced to 14 men before the break when Naiyaravoro was yellow carded for a no arms tackle on Warren.
However, despite being a man down, the home side struck on the stroke of half-time as hooker Fish burrowed over from a driving maul.
Dragons did hit back seconds after the restart as Amos kicked in behind and Warren picked up the beautifully waited ball and burst through a tackle to race over and cut the lead.
But within two minutes Saints had secured the try bonus as Burrell hit the line late and at pace to crash in under the posts.
Big wing Naiyaravoro used all his power to slam over the whitewash before Saints scored the try of the game as slick handling saw Teimana Harrison and Naiyaravoro combine on halfway and feed Mitchell to race clear.
Naiyaravoro added his second of the game out wide as Dragons were forced to defend for long periods under huge pressure.
Dragons looked to have the final word as Cudd went over from a driving maul.
But back came Saints with Sleightholme going over with the final play of the game.
02m Burrell t 5-0
11m Ratuniyarawa t Biggar c 12-0
39m Fish t Biggar c 19-0
42m Warren t Lewis c 19-7
46m Burrell t Biggar c 26-7
51m Naiyaravoro t 31-7
53m Mitchell t 36-7
58m Naiyaravoro t 41-7
78m Cudd t Lewis c 41-14
79m Sleightholme t Biggar c 48-14
Referee: Thomas Charabas (Fra)