Jones finished with a 20-point haul - four penalties, two drop goals and a conversion.
The winner came after Wasps' former Scotland full-back Hugo Southwell was sin-binned for obstructing Dragons winger Ifan Evans as he set off to chase his own kick towards the Wasps line.
Earlier, full-back Hallam Amos celebrated becoming the youngest Welsh player to appear in regional rugby with a 31st-minute try .
But Amos - 17 years and 28 days old - lost the record two minutes later when Jack Dixon - even younger at 16 years and 333 days - took the field for a seven-minute cameo appearance as a blood replacement while left-wing Michael Poole had a head wound treated.
Dragons' other try came from prop Phil Price.Wasps replied with tries from skipper John Hart, winger Tom Varndell, former New Zealand lock Ross Filipo and scrum-half Joe Simpson, on as a replacement after returning from England World Cup duty, with fly-half Ryan Davis kicking two conversions and a penalty and his replacement Nicky Robinson also converting.
Price struck after only three minutes, forcing his way over following an effective rolling maul.
Davis opened Wasps' account four minutes later with his first penalty after Dragons pulled a scrum down.
Playing the advantage after a Wasps infringement, Jones put the Welsh side 8-3 ahead with a superbly-struck 14th minute drop goal from only two metres inside the Wasps half, before landing a penalty three minutes later.
Hart, who produced an inspirational captain's performance, rounded off a move involving Elliot Daly and Filipo, with his side's opening try after 20 minutes, with Davis converting.
Jones responded with his second penalty before centre Chris Bell's one-handed reverse pass switched the direction of the Wasps attack for Hart to deliver the pass which sent former England winger Varndell racing in.
Dragons took only two minutes to move back in front, 21-17, with Amos' converted try but Wasps squandered a gilt-edged opportunity to go back in front before half-time.
Winger Christian Wade - the Aviva Premiership's leading try-scorer with six from six matches this season - went over in the right-hand corner but the score was disallowed because Varndell's final pass was forward.
Filipo, signed from French club Bayonne in the summer, collected his first Wasps try three minutes into the second half when the home side were allowed to play the advantage after Evans denied them a try on the left by deliberately knocking on.
Wasps moved the ball the full width of the pitch before hooker Tom Lindsay and prop Jeremy Castex, signed in the close season from Narbonne, provided the final two passes for Filipo to go over in the right corner.
Lindsay and Castex were then replaced by Rob Webber and Samoan prop Zak Taulafo, back from the World Cup.
Jones booted the Dragons back into a 24-22 lead with a penalty from the halfway line.
Wasps responded by introducing Simpson and centre Dominic Waldouck, making his 100th appearance for the club - and his first since undergoing Achilles tendon surgery six months ago.
Robinson, the English side's Welsh replacement on at fly-half for the injured Davis, shaved the outside of the post with a penalty and then missed a drop-goal attempt.
Davis landed his second drop goal for the Dragons six minutes from time.
But, with three minutes to go, Simpson darted over following a Wasps scrum and Robinson converted to give the home side a 29-27 lead - only for their celebrations to prove premature as the immaculate Jones had the final say.
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