Jefte was ejected from the game in a harsh manner by the referee as two rapid Dynamo Kyiv goals crushed Rangers' Champions League dreams for yet another season.
Cyriel Desser's last-gasp equaliser in the first leg had given them hope and both he and Ross McCausland had chances to give them the lead early in the Hampden tie.
Already on a booking, Jefte was punished in the 50th minute with a yellow as he jumped to win a header and was shown a red card that made it a struggle for the hosts.
They held on for 30 minutes but Philippe Clement's side eventually collapsed and substitutes Oleksandr Pikhalyonok and Nazar Voloshyn scored in quick succession to send Dynamo through to the play-off to face RB Salzburg.
Rangers drop into the Europa League group stages for another season.
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