Contepomi was the man of the match even though Eddie O'Sullivan's side managed to control the Pumas captain in a first half which belied Ireland's inexperience at full-back.
Simon Best, in his first match as captain, did not disappoint with an outstanding performance.
Full-back Gavin Duffy and wing Brian Carney provided a telling influence in the early part of the game, Carney's movement surprising the hosts as the former rugby league player constantly probed Argentina's defence.
And it was Ireland who broke the deadlock with a Paddy Wallace penalty after six minutes.
A minute later Carney had his reward when he recovered the ball after a misplaced pass by Contepomi and claimed his first try for Ireland on his international debut.
Wallace slotted the conversion to put the visitors into a 10-0 lead. Â
Contepomi's penalty then reduced the deficit and after he and Wallace missed further kicks at goal, the Argentinian converted just before the break.
Ireland made a nervous start to the second half before Contepomi reduced the deficit to a single point.
Duffy's great kick then gave Ireland great field position and a fantastic 30-yard rolling maul eventually saw the visitors awarded a penalty try. Wallace's conversion put them 17-9 ahead.
However, Argentina hit back through a Contepomi penalty before Hernan Senillosa went over for a try. Contepomi's conversion put the hosts ahead for the first time in the match.
Although Duffy's penalty put Ireland ahead again with three minutes to go, Contepomi's last-gasp drop goal saw Argentina take the lead.
Victory was assured for the hosts when wing Geordan Murphy failed with an attempt at a drop goal.