Divij, who made his Grand Slam debut at Wimbledon, and Karlovic got the better of the Ukrainian pair of Illya Marchenko and Denys Molchanov 4-6 7-6(1) 11-9 in the opening round of the USD 455,775 grass court event.
Karlovic is playing his first tournament after recovering from viral meningitis.
In the singles, Prakash Amritraj suffered mauling at the hands of former world number one Lleyton Hewitt, losing his second round match 2-6 1-6. The Australian broke Amritraj six times while saving seven of the eight chances he faced.
Divij was broken in the fifth game at deuce. Both teams went on to hold serve after that and the Ukrainians pocketed the set, riding on that break.
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The Indo-Croatian again squandered a break chance in the 11th game of the second set, which was eventually decided via tie-breaker. Divij and Karlovic raced to a 6-0 lead before winning the tie break 7-1.
In the match tie break, Divij and Karlovic were staring down the barrel, down 4-9, but saved five match points and won seven points in a row to win 11-9.