It was a low-scoring encounter and that was down to a tremendous defensive shift from Mike Ruddock's team.
They led 7-6 at the break despite spending most of the half on the back foot due to two first half sin binnings to Will Bowley and Pat Sanderson.
The Dragons were physical opponents and gave the Warriors a stern and physical defensive workout, however, it was the visitors who were first on the scoreboard when Alex Grove back from international camp with Scotland intercepted a loose Wayne Evans pass inside his own 22 with Warriors fly-half Willie Walker finished off after Latham was inches short before converting his own try
Early seccond half posession saw Harries cross the line following a strong break by replacement flanker James Harris.
After Warriors failed to make the most of good posession deep inside the Dragons' 22, their misery was compounded when Connor kicked over an injury-time penalty from 50 yards to secure the win for the Welsh region.
Worcester Warriors 10 - 16 Newport Gwent Dragons (Rodney Parade, Newport) - 21 August 2009
7 mins: Willie Walker Try 0-5; 8 mins: Willie Walker Conversion 0-7; 10 mins: James Arlidge Penalty 3-7; 13 mins: James Arlidge Penalty 6-7; Half Time 55 mins: Willie Walker Penalty 6-10; 73 mins: Will Harries Try 11-10; 74 mins: Shaun Connor Conversion 13-10; 84 mins: Shaun Connor Penalty 16-10
Adam Black, Aleki Lutui, Jack Gilding, Will Bowley, Craig Gillies, Tom Wood, Pat Sanderson (capt), James Rodwell, Johnny Arr, Willie Walker, Miles Benjamin, Greg King, Alex Grove, Marcel Garvey, Chris Latham
Replacements: Sam Tuitupou for Greg King (35), Calum MacRae for Sam Tuitupou (44), Greg Rawlinson for James Rodwell (40), James Collins for Will Bowley (47), Olivier Sourgens for Jack Gilding (48), Chris Fortey for Aleki Lutui (59), Ryan Powell for Johnny Arr (59), Matt Cox for James Collins (68)
Try: Willie Walker Con: Willie Walker Pen: Willie Walker
Cards: 13: Will Bowley - Yellow, 33: Pat Sanderson - Yellow
Newport Gwent Dragons
Phil Price, Tom Willis (capt), Ben Castle, Adam Jones, Luke Charteris, Dan Lydiate, Gavin Thomas, Lewis Evans, Wayne Evans, James Arlidge, Richard Fussell, Ashley Smith, Tom Riley, Aled Brew, Jason Tovey
Replacements: Ali McKenzie for Phil Price (35), Steve Jones for Tom Willis (40), Nigel Hall for Ben Castle (40), Robert Sidoli for Luke Charteris (40), Grant Webb for Lewis Evans (40), James Harris for Gavin Thomas (40), Martyn Thomas for Jason Tovey (40), Will Harries for Richard Fussell (40), Alex Walker for Wayne Evans (40), Shaun Connor for James Arlidge (40), Richard Fussell for Aled Brew (75)
Try: Will Harries Con: Shaun Connor Pen: James Arlidge (2), Shaun Connor
Referee: Gwyn Morris (Wales)