Djere, ranked 86, had lost both his earlier Davis Cup matches before clashing with Ramkumar, ranked 135, at Kraljevo Sports Venue.
Having got first-round byes yesterday, Indians had a busy day with matches in both singles and doubles.
Ramkumar Ramanathan had said he would love to win a medal with Leander Paes at the Asian Games, beginning August 18 in Jakarta and Palembang.
Ramkumar, who ended a runner-up at the Newport event last week, lost 4-6, 4-6 to American world number five Taylor Fritz.
The 23-year-old from Chennai beat the American 6-4 7-5 to become the first Indian in seven years to make it to the singles title round on ATP World Tour.
The world number 161 has made his first ever entry into the last four of any ATP tour.
Leander Paes bowed out after losing his doubles quarter-final at the Hall of Fame Open grasscourt tournament in Newport.
Yuki, who qualified for the Indian Wells Masters and went on to scalp world number 12 Lucas Puille, dispatched Argentina's Olivo 6-4 6-1 in the opening round of the USD 7,972,535 hard court event.
In his opening round in the main draw, Yuki will fight it out with fellow qualifier Nicolas Mahut, ranked 101, and if he manages to cross the first hurdle he will be up against world number 12 Lucas Pouille.
Yuki, seeded ninth, edged past America's Dennis Novikov 7-5, 7-5 while Ramkumar defeated Michael Mmoh 6-3, 7-5 in his first-round match in the qualifiers on Wednesday.
While Cilic will be the outright favourite in the next match, Ramkumar had a confidence-boosting win over world number eight Dominic Thiem at Antalya Open last season.
The Indian tennis legend also stressed on the fact that we need good singles players if India want to do well in Davis Cup World Group as the top players take part and have to look beyond only doubles events.
Yuki, country's top singles player, overcame Ramanathan 4-6 6-3 6-4 in a little over two hours at Balewadi Sports Complex.
With the outcome already decided, Yuki Bhambri won the dead fifth rubber against Canada's Brayden Schnur 6-4 4-6 6-4 as India lost the World Group Play-off tie 2-3 to the hosts.
This was Ramanathan's second consecutive defeat against the American, having lost to him in the second round of the Roland Garros Qualifier in 2015.
After going down in the first set, Ramkumar Ramanathan rebounded strongly to register a 6-7 (5-7), 6-1, 6-4 win over Christopher Eubanks .
The 22-year-old from Chennai is now India's best-ranked player in the singles.
Ramkumar Ramanathan will next face veteran Cyprus star Marcos Baghdatis in the quarter-final.
India today took an unassailable 3—1 lead over New Zealand after Ramukumar Ramanathan routed Finn Tearney.
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