The Australian Open champions outclassed their Dutch opponents 7-5, 6-4 win in an hour and 11 minutes, despite rain interruptions.
The second-seeded pair showed great athleticism on the court to keep their opponents at bay in the opening encounter of the grass court major.
Bopanna and Eden will take on the German pair of Hendrik Jebens and Constantin Frantzen in the second round on Thursday.
However, Sumit Nagal's campaign ended at a low after losing his first-round match of the men's doubles with his Serbian partner Dusan Lajovic on Wednesday. The duo were defeated by Spain's Pedro Martinez and Jaume Munar by 6-2, 6-2.
Earlier, Nagal faced a first-round exit in the men's singles in his maiden main draw appearance at Wimbledon.
Meanwhile, N Sriram Balaji and Yuki Bhambri will play first-round men's doubles match against fourth seed Serbian-Salvadoran pair of Mate Pavic and MarceloArevalo later in the day.