Smith, 27, is in his ninth season with the region and has clocked up 145 appearances for Lyn Jones' side.
"Ashley is a Dragons stalwart just like (flanker) Lewis Evans," said Jones.
"They are players that give 100 per cent to the region every time that they put the shirt on and we need more people like them in our setup."
"Tom has shown great promise throughout his time with the region and I am looking forward to working with him over the coming seasons.
"He delivers and responds to everything the coaches and I throw at him, he has great potential as a player and I hope that he realises that himself."
Commenting on his renewal, Smith said, ""I am delighted to have been able to re-sign for another three years with my home town club.
"Under Lyn's guidance the club is going places and will become very successful in the coming seasons. I look forward to what the future holds with the Dragons and being part of its success."
Prydie added, "I am pleased that I have re-signed for the Newport Gwent Dragons. I am really looking forward to being here over the next three seasons alongside the rest of the squad."
The Dragons travel to Llanelli on Friday May 2, to face the Scarlets in their penultimate game before hosting Benetton Treviso at Rodney Parade on Saturday May 10.