The British pair squeezed past their opponents 2-6, 7-6, 11-9 and into the second round after saving five match points, delaying Murray from ending his glittering career.
Murray, the first male tennis player to claim two Olympic singles gold medals and widely regarded as one of Britain's all-time great sportsmen, said he would not compete in the singles to focus on doubles in his final tournament before retirement.
After losing the first set and going down 4-2 in the second, the British pair stormed back thanks to a strong net game, pushing the Japanese pair to a tiebreak to win the second set.
Nishikori and Daniel quickly took the lead in the match tiebreak but the Britons hit back, saving five match points before closing the contest out in a little over two hours.
