The Saracens star has been a doubt for the clash in Hamilton following last weekend's narrow 28-17 defeat to the All Blacks.
It has now been confirmed by the team that Farrell has been unable to recover in time to end the series, which New Zealand clinched in Dunedin.
Head coach Stuart Lancaster said: "The knee has been scanned. It's disappointing for Owen as he was desperate to play in the final Test.
"Such is his character that he is already working on his rehab and still contributing to the tour. Fortunately we have three other fly halves and we'll make that selection call over the next few days."