The Indians, making their debut as a pair in the elite season-ending event, lost their Group B match 6-4 7-6 (1) 10-12 against the British-Denmark combination in one hour and 36 minutes.
Bhupathi and Bopanna are coming into the top-8 teams event after winning the Paris Masters last week.
The fifth seed Indian pair will now have to win the last two Group B matches to keep themselves in the semi-finals hunt.
Bhupathi and Bopanna came back well after losing the first set and even had two match points in the Super Tie breaker but could not cash in on.
Leander Paes and his Czech partner Radek Stepanek, seeded third in Group A, will open their campaign against Paris Masters finalists and seventh seeds Aisam-ul-Haq Qureshi of Pakistan and Dutch Jean-Julien Rojer.