Back then Peru, leading 2-1 in Buenos Aires, were nine minutes from the win they needed to qualify when Gareca scrambled a late equaliser for the hosts.
Argentina qualified instead and won the tournament, although Gareca was left at home by coach Carlos Bilardo.
Peru were forced into a playoff and lost to Chile, ending an era which had seen them reach three out of four World Cups and starting a run of eight straight failures to reach the finals.
Having endured a 36-year absence, Peru are finally back for next year's tournament in Russia with a team which, in one of soccer's great ironies, is managed by Gareca.
After coaching clubs around South America, Gareca was offered the Peru job in 2015 and made an instant impact, leading them to the semi-finals of that year's Copa America in Chile.
That feel-good factor was taken into the World Cup qualifiers and, after a slow start, they lost only one of their last eight games, finishing in fifth place before beating New Zealand in an intercontinental playoff.
It was a remarkable turnaround, achieved without any big name players.
Only one of the 23-man squad which faced New Zealand is based with a club in one of the so-called big five leagues of Europe - winger Andre Carillo who plays for Watford.
Others are scattered around lesser European leagues such as Denmark, Russia, Portugal and the Netherlands, with others in Mexico, Brazil and Ecuador and a surprising number plying their trade in Peru's cash-strapped domestic league.
But despite apparently limited resources, Gareca has transformed the team, ridding them of the fear of failure which had been gnawing away over the years.
He has restored Peru's ability to pass their way through an opposing midfield and built a side who are adept at winning the second ball and dangerous on the counter-attack.
Even without captain Paolo Guerrero, who failed a doping test after the match against Argentina and could receive a ban that would keep him out of the tournament, they are likely to be a handful for any opponent.
And if Peru's players are not well known to neutrals, their kit certainly is -- the fanous white shirt with a large diagonal red stripe is one of the most iconic in international football.
(Writing by Brian Homewood; editing by Ken Ferris)