The Men of Gwent made a good start and led 3-0 thanks to Tovey booting over a penalty, earned from a scrum. But Scarlets replied with two penalties from Rhys Priestland and a converted try by Gareth Owen making it 13-3.
The Dragons centre Ben John then suffered an injury to his hamstring, which meant that Tom Prydie, who had taken Lee Byrnes place as fullback, slotted in next to Ashley Smith in the centre and Geraint Rhys Jones came on as fullback, with Brew and Amos covering the wings.
The Scarlets constant attacking against the visitors gained them 3 points as Priestland kicked another penalty to make it 16-3 yet Tovey responded with another penalty earned at the set piece.
However, the game was secured when the Dragons spilt the ball in midfield, Owen hacked on and wing Kristian Phillips won the race with Tovey to touchdown.
Priestland converted to make it 23-6 before Dragons scrum-half Richie Rees scored a consolation after an impressive driving lineout by the visitors.
Full Time: Scarlets 23 - Dragons 6
Scarlets: Liam Williams, Harry Robinson, Gareth Owen, Scott Williams (c), Kristian Phillips, Rhys Priestland, Gareth Davies; Rob Evans, Emyr Phillips, Samson Lee, Jake Ball, Joe Snyman, Aaron Shingler, John Barclay, Rory Pitman.
Replacements: Kirby Myhill, Phil John, Rhodri Jones, George Earle, James Davies, Aled Davies, Steven Shingler, Michael Tagicakibau.
Newport Gwent Dragons: Tom Prydie, Aled Brew, Ben John, Ashley Smith, Hallam Amos, Jason Tovey, Jonathan Evans; Boris Stankovich, T. Rhys Thomas, Dan Way, Ian Gough, Rynard Landman (c), Andy Powell, Nic Cudd, Taulupe Faletau.
Replacements: Elliot Dee, Hugh Gustafson, Lloyd Fairbrother, James Thomas, Lewis Evans, Richie Rees, Angus O'Brien, Geraint Rhys Jones.
Man of the Match: Liam Williams (Scarlets)
Referee: George Clancy (Ireland)