Spaniard Robredo lifted the title in Bastad in 2006 and 2008 but he fell to a 7-6 (8-6) 6-3 defeat at the ATP 250 clay-court event.
World number 258 Olivo, making his Tour debut in Sweden, started strongly with an early break of serve only to allow Robredo back into the set as the momentum ebbed and flowed in the match.
Having forced a tie-break, Robredo lost his serve twice to allow Renzo to move ahead, and the young Argentinian carried that into the second.
Robredo's serve then began to let him down dramatically as he only managed to get 47 per cent of his first attempts in, and Olivo was pouncing on the second winning 63 per cent of the points.
The world number 18 still had a chance to level the match after a break in the fifth game, but he lost four games in a row to tumble out of the competition.
Another seed out of the tournament is Jerzy Janowicz after the Pole was forced to retire with foot pain at 6-3 1-1 down to Dusan Lajovic.
Joao Sousa, the fifth seed, had no such problems in his match against Sweden's Elias Ymer as he raced to a 6-3 7-5 win in 88 minutes.
Meanwhile, the final Swedish representative in the competition saw his run end in the second round as wildcard Christian Lindell fell to a 4-6 6-2 7-6 (7-2) defeat to Pablo Cuevas.