Rangers started on the front foot, with the Dynamo defence looking nervous and hesitant.
Jefte was lively in his first Rangers start, heading wide at the back post from a James Tavernier cross before Cyriel Dessers nodded straight at goalkeeper Heorhiy Bushchan.
Bushchan had to get down quickly to push away a low Ross McCausland strike, while Dynamo were restricted to a couple of dangerous Vladyslav Kabayev crosses flashed across the Rangers six-yard box.
Rangers manager Philippe Clement decided to replace McCausland with Vaclav Cerny at half-time, but he might have been left wondering if he should have made a change on the other wing when Jefte got his marching orders.
The decision from Italian referee Marco Guida appeared harsh on the 20-year-old, but the summer arrival from Fluminense's participation was always on a knife edge after an early first yellow.
As the possession statistics tipped in favour of the visitors, Ukraine midfielder Karavayev dragged a shot wide on the break.
Rangers applied some intense pressure of their own despite being down to 10 men but were undone when midfielders Pikhalyonok and Voloshyn exploited the growing spaces left by the hosts.