The Gers were left to rue the second half decision from referee Marco Guida to dismiss Jefte for a perceived elbow, despite replays showing that the Brazilian had merely outjumped his opposite number.
And despite a valiant effort from Philippe Clement's side with the tie still firmly poised at 1-1 on aggregate, the away side eventually made their numerical advantage count through late goals from Oleksandr Pikhalonok and Nazar Voloshyn.
It means Rangers will drop into the UEFA Europa League group stages, with the draw set to be made later this month.
The Gers lined up with two changes from the side which defeated Motherwell at the weekend, with Clement handing starts to Ross McCausland and Jefte in place of Scott Wright and Vaclav Cerny.
There was an air of a classic European qualifier in Mount Florida as the sun was yet to set with the teams lined up for kick-off.
And it was the visitors who engineered the first opportunity of the contest, albeit of the half-chance variety, when Vladyslav Kabaiev advanced down the left-hand side and centred in search of last week's goalscorer, Andriy Yarmolenko, but the cross whizzed past the former West Ham United man at full stretch.
Rangers took some time to find their footing in the match, with Jefte producing the first effort in anger for Clement's side when he rose highest to meet a deep delivery from James Tavernier before powering his header down into the turf and wide of the target.
The Brazilian summer signing was the Gers' brightest spark in the opening blows, going into the referee's notebook for a late challenge on the half-hour mark minutes after a pinpoint delivery found the head of Cyriel Dessers who nodded straight into the grateful arms of the Dynamo Kiev goalkeeper.
It spoke volumes about the cageyness of the contest, teed up perfectly for the neutral after the 1-1 draw in Lublin seven days ago, that Rangers' most threatening moment of the first half arrived on the counter-attack.
From a Dynamo Kiev free-kick, Dessers peeled smartly into space before retaining possession long enough to allow Ross McCausland to hare into open space. The Northern Irishman shifted the ball onto his favoured right side on the edge of the penalty area, but his effort was of the tame variety and Georgiy Bushchan got down smartly to save.
The half-time interval prompted the introduction of Vaclav Cerny, who announced himself to the Rangers faithful on Saturday with a spectacular strike against Motherwell.
However, Rangers were reduced to ten men just four minutes after the restart when Jefte was shown a second yellow card for contesting an aerial challenge.
On the touchline, Clement looked completely perplexed by the decision, and a quick look at the action replay showed that the Gers boss was justified given the innocuous - and unavoidable - nature of the contact from the Brazilian.
The decision certainly ignited the Rangers support, although the numerical advantage in favour of Dynamo Kiev almost paid dividends immediately when Oleksandr Karavaiev raced clear of the Gers defence in transition but could only drag his effort wide of Jack Butland's goal.
Taras Mykhavko was next to test his luck from distance for the away side, although his rising effort from all of 30 yards sailed a long way over the Rangers target.
The numerical disadvantage never appeared to spook Rangers' attacking endeavour, underlined with 18 minutes remaining of normal time when Cerny did tremendously to weave his way to the byline and pull the ball back for Dessers to strike against a desperate Dynamo Kiev block.
However, the Ukrainian side made their superiority in numbers count with nine minutes remaining when substitute Pikhalonok received a pass inside from Karavaiev and rifled an unstoppable low finish into the bottom corner.
One soon became two against the ten men of Rangers less than 60 seconds later following a neat combination between Mykola Shaparenko and substitute Voloshyn on the left-hand side which allowed the latter to finish smartly under Butland and extinguish any lingering Gers hopes.
RANGERS: Butland, Tavernier, Souttar, Propper, Ridvan (Davies, 76), Barron, Diomande (Sterling, 66), Lawrence (Matondo, 83), McCausland (Cerny, 45), Jefte, Dessers
SUBS: Kelly, Dowell, Wright, King, Fraser, McKinnon, Fraser, Danilo
DYNAMO KYIV: Bushchan, Popov, Brazhko, Yarmolenko (Voloshyn, 66), Shaparenko (Lonwijk, 89), Vanat, Karavaiev (Pikhalonok, 74), Kabaiev (Bilovar, 89), Buyalskyi (Braharu, 74), Mykhavko, Dubinchak
SUBS: Neshcheret, Vivcharenko, Andriyevskiy, Supryaha, Tymchyk, Ceballos, Diallo
REFEREE: Marco Guida (ITA)