In contrast the Dragons came close to ending their woeful away record but in the end came up short and they have now just won once on the road since last October.
The Gunners had thebetter of the opening exchanges but a well-organised Dragons defence ensured they had little reward for their efforts.
The pair snapped up their all-important late touchdowns after they had looked destined to pay dearly for firing blanks in attack.
With less than 10 minutes left, Todd Blackadder's troops were trailing and looked unlikely to pull off a salvage act, but Argentinian speed merchant Leonelli and winger Webster had other ideas - and skipper Chris Paterson's
conversions made sure of the win.
Chris Paterson gave the hosts the lead with an easy penalty but the scores were soon levelled when Craig Warlow slotted over.
The Gunners continued to struggle to capitalise on their dominance but new recruit Leonelli impressedon more than one occasion.
After the break the Gunners still lacked a clinical finish, with Simon Tayor wasting an opportunity, and they were lucky not to be punished when Warlow sent a penalty wide.
It was left to the visitors to show how to finish when just after the hour mark lock Luke Charteris made the initial break before Gareth Wyatt put centre Hal Luscombe in at the corner.
But that appeared to spark the Gunners into action with Leonelli getting due reward for his performance.
The speedster began and finished the move, that also saw key contributions from Hugo Southwell (twice) and Paterson, who also added the conversion.
Then in the dying moments Webster grabbed another try that Paterson also converted.
But their was some consolation for the Dragons when a last-gasp penalty from Warlow ensured they went home with a bonus point.
|Edinburgh Gunners Score Card|
|Francisco Leonelli Morey||1||5|
|Newport Gwent Dragons Score Card|