Jefte was harshly sent off for Rangers before two quick Dynamo Kyiv goals ended their Champions League hopes for another season.
Cyriel Desser's last-gasp equaliser in the first leg had given Rangers hope and both he and Ross McCausland had chances to give them the lead early in the second leg of the third qualifying round tie at Hampden Park.
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 seal a 2-0 win on the night - 3-1 on aggregate - to send Dynamo through to the play-off to face RB Salzburg.
Heartbreak at Hampden
It was not a great night for Clement's side who will see this as an opportunity lost as they drop into the less prestigious and certainly less lucrative Europa League group phase.
Rangers made a high-pressing start and the visitors initially looked uneasy in defence.
However, Oleksandr Shovkovskyi's side worked their way into the game and in the ninth minute winger Vladyslav Kabaev sped past Rangers captain James Tavernier down the left but his inviting cross found no takers, to Rangers' relief.
Rangers made little headway until the 20th minute when Jefte nodded a looping Tavernier cross past the far post, before thundering a snatched shot from 25 yards high over the bar minutes later.
A Dessers header from a Jefte cross was easily saved by Kyiv 'keeper Georgiy Bushchan and then, as Rangers pressed, the ball fell to Tom Lawrence inside the box but he could not sort out his feet to get a shot away.
When they did attack, Kyiv looked pacey and purposeful which kept Rangers' defence on a state of high alert.
In a Rangers counter just after the half-hour mark, Dessers played in McCausland but although his shot was parried by Bushchan, Clement's side failed to take advantage of the loose ball.
Cerny replaced McCausland for the start of the second half which gave Rangers fans reason for optimism but that disappeared when Jefte, booked for a touchline foul on Andriy Yarmolenko after 30 minutes, was cautioned again and sent packing for an aerial challenge on Oleksandr Karavaiev.
When the dust had settled on what looked like a soft second yellow, and the Rangers fans cranked up the noise in defiance, Karavaiev raced clear of the defence but screwed his shot wide.
Kyiv took advantage of the extra player and moved with menace before Rangers rallied, and the game swung end to end.
In the 68th minute Dujon Sterling replaced tiring midfielder Mohamed Diomande and Ben Davies took over from Ridvan Yilmaz, who was taken from the field on a stretcher after picking up an injury after taking a shot minutes beforehand.
It was another blow for the Light Blues and Dessers seemed to take umbrage with the fans after hearing groans and moans when he lost possession when he had no support.
Rangers 'keeper Jack Butland was hardly under fire but he did save a long distance effort from Taras Mykhavko.
However, in the 82nd minute, he had little chance when Pikhalyonok fired in from distance under no pressure from the Rangers defence.
Voloshyn then converted a cut-back from Mykola Shaparenko to seal the tie, with Butland rescuing himself at the end when he saved a shot from Vladyslav Vanat, after his initial mistake.
Clement: The worst decision I have seen
Rangers manager Philippe Clement on the red-card decision:
"I saw images and it is clear in football that sometimes it is a grey zone and because of that VAR is there and it helps to make football more honest.
"But the decision was nothing to do with grey zone. It was really clear, there was nothing going on.
"Jefte jumps higher, he doesn't move his hand, his arm is next to his body, it is nothing.
"I try to understand the decision and ask but the referee stuck to his opinion, it was a clear foul and a second yellow and a red. I am confident his bosses will have another idea.
"It is a very decisive moment and in the end it has killed the dream of the dressing room and killed the dream of more than 50,000 fans and you expect a better level of decision-making.
"This is the worst decision I have seen in more than 30 years of football.
"I have a lot of things in my head but you guys know if I say too much I can get a ban so let's keep things in my head.
"I need to be good and smart for the club and the team and we need to move on and take positive lessons from the two games."
Boyd: A hammer blow
Sky Sports Kris Boyd on Jefte's sending off:
"How that's given as a second yellow card is beyond me. He jumps up with his arms by his side, the Dynamo Kyiv player goes to the ground and the referee buys it and gives him a yellow card.
"I know with VAR you can't go and check a second yellow card but it was horrendous.
"You've got to pick yourself up from that but you got the feeling Rangers could have played all night and not scored.
"This is a hammer blow for Rangers. I'm not saying they would've beaten Salzburg in the next round but the £5m would have been huge, could that have been invested in the squad to strengthen?
"It's a lot of money for Rangers not to get, especially when they go to Poland last week and the performance they put in.
"Also, 39,000 at Hampden. Rangers fans are not happy and rightly so. This whole situation with the stadium has come back to cost Rangers again.
"Champions League nights at Ibrox, teams crumble. Tonight there wasn't much of an atmosphere at Hampden."
What's coming up for Rangers?