The Pumas, shorn of up to a dozen first team regulars, exploited a slow start from Clive Woodward's men to race 13-0 ahead after just 15 minutes as Jose Nunez Piossek burst over for a try in the corner.
The Pumas then held on as fly-half Federico Todeschini enjoyed a perfect evening with the boot.
Ollie Smith scampered over for the Lions but Todeschini's 20 point haul had seemed enough to keep the opposition caged before Wilkinson snatched the draw with his seventh successful kick from seven attempts.
Argentina drew first blood, pushing the Lions deep inside their own 22 and when Lewis Moody was penalised for flying into a ruck from the side, Todeschini slotted over his first kick.
Then with just six minutes gone, Felipe Contepomi picked up a loose ball and sliced away on a sharp break before feeding Piossek in space down Shane Williams' wing for the opening try.
Todeschini converted and after Williams briefly dazzled, Smith cut through onto a Wilkinson pass but was penalised for holding on and Todeschini slotted an excellent penalty.
Two minutes later the Lions showed their first real flash of invention and it earned them a try.
Wilkinson took the ball up to the line and slipped a perfectly timed ball inside to Smith, who burst through and dived over to put the Lions on the board.
The Pumas continued to pound away, pressing deep into Lions territory and the pressure was only relieved when Michael Owen, the youngest Lions captain since John Spencer in 1971, emerged with ball and broke clear of his own line.
Williams sparked into life as he counter-attacked under a poor Argentina clearance but his deft chip ahead was smothered and ushered into touch.
The Lions pressed for a second try before half-time and began to find their feet but were stymied by an Argentina defence camped offside.
Either side of another long-range effort from Todeschini, Wilkinson booted two more penalties, both of them for offside.
Geordan Murphy wasted a promising break down the left with a sloppy pass but the Lions eventually drew level as Wilkinson slotted his fourth penalty.
But not for long. Todeschini nudged Argentina ahead with two quick penalties, the second awarded after the Lions had been sent reeling by a 30-metre rolling maul.
Wilkinson, though, was beginning to impose his authority on the game.
He looked assured going forward and fired a wonderful bullet pass out to Murphy that almost broke the Argentina defence.
Wilkinson then darted himself and after being felled by a high tackle slotted the penalty that brought the Lions to within three points.
Camped in the Argentina 22 and urged on by Neil Back, acting as a water carrier, the Lions continued to kick penalties to touch in search of the winning try.
In the fifth minute of injury time, the Lions forced a mistake at the scrum-base, won possession but Gordon D'Arcy dropped the ball with a clear run for the line.
Williams then jinked inside but the Argentina defence clung on, forcing the Lions to abandon any hope of victory as Wilkinson slotted the kick that ensured honours ended even.