The 27-year-old was unable to inspire the Bundesliga champions as they slumped to a 3-1 second leg loss at the Allianz Arena.
Sadio Mane scored in either half as the Reds joined Premier League rivals Manchester City, Manchester United and Tottenham Hotspur in the draw for the quarter-finals.
Rodriguez was on the field for 74 minutes but was unable to make a tangible contribution.
He completed 43 of his 49 passes during the 90 minutes, a success rate of 88 per cent.
But there was little else to show from the Real Madrid loanee's display.
The future of the attacking midfielder is uncertain as he approaches the end of his two-year loan spell at Bayern.
Already confirmed to return to Real at the end of the season, the return of Zinedine Zidane is unlikely to improve his opportunities at the Bernabeu.
Arsenal were linked with a loan move for Rodriguez in January but the deal never materialised.
Since winning the World Cup golden boot in 2014, which earned him a £63million move to Real, Rodriguez has failed to to recapture his best form.
But after his most recent display, Arsenal fans were left thrilled that the January move never went through.
Twitter user Don Unai said: "James Rodriguez is a certified bum! Fraud his entire career on that 2014 World Cup performance. Levante, Getafe level player."
Tim Richards added: "James Rodriguez has no right foot. Fine, some people are very one footed. Arjen Robben, Lionel Messi… the issue here is that they offer a great deal more in terms of speed and agility.
"When James gets the ball playing in the middle of the park, he can only move one way."
Mocking the idea that Rodriguez could have replaced Mesut Ozil, who was also linked with a January move Sidney said: "Some Arsenal fans wanted James Rodriguez to replace Mesut Ozil…LMAO."
Smokey stated: "James Rodriguez s***e again. Wouldn’t want him at Arsenal, such a weak player mentally.
"He’s still living off one good game in the World Cup."
Tottenham supporter Jason Gish summed up The Emirates Stadium club's recent transfer success by writing: "James Rodriguez is massively overrated. I look forward to him replacing another overrated midfielder at Arsenal."
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