The Indian pair, seeded fifth, were making their first appearance together at the season-ending event, although Bhupathi had qualified on 11 previous occasions with three different partners, and they will have to win their last two matches to keep alive their hopes of reaching the semi-finals.
After Marray and Nielsen took the first set, Bhupathi and Bopanna hit back in the second to set up a super tie-break to decide the match.
Bhupathi and Bopanna should have won it, but their opponents managed to save two match points and then converted their first match point to steal the Group B victory.