With home advantage, Lyon took the lead as early as the ninth minute through Jimmy Briand, but were pegged back in dramatic fashion by Pierrick Valdivia's penalty in second-half stoppage time.
Four minutes into the extra period, Jean-Philippe Gbamin headed home from a corner to stun Lyon, who last week earned a place in the final of the Coupe de la Ligue, where they will face Paris Saint-Germain.
The result left Garde frustrated with the way his side had allowed their opponents back into the game, with Lens progressing to a last-eight tie against Monaco.
"We are disappointed to have missed qualifying," he said.
"Having opened the scoring and having some control over the game, we squandered too many chances.
"We did not put them away and we were punished.
"When you have a lot of opportunities in a game but you keep a one-goal lead, it gives the opponent an opportunity."
Lyon have a chance to bounce back when they host Ajaccio in Ligue 1 on Sunday.
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