Playing under the roof at Ariake Colosseum because of rain, del Potro came back from a sluggish first set to secure the win in just over two hours.
"The match came to me in the end," del Potro said. "I was focusing on breaking his serve and finally did it. He was hitting a lot of aces (17)."
His opponent in the second round will be Argentine compatriot Carlos Berlocq.
Nishikori was just two points from defeat while trailing 5-2 in the second set, but reeled off five straight games.
Del Potro, ranked seventh in the world, prevailed with solid play from the baseline and capitalized on some crucial mistakes by Baghdatis.
The crucial point in the match came in the eighth game of the final set, when del Potro converted on his fourth break point chance to go up 5-3 after Baghdatis hit a return into the net.
"It was important for me to win this kind of match," del Potro said. "Playing him in the first round is probably the worst person to face. He is unseeded here and dangerous. He plays well on the hard-courts."
Del Potro was a late addition to the tournament after Andy Murray, the 2011 champion, pulled out due to a back injury. Del Potro is seeking his third title of the season, following victories in Rotterdam and Washington.
In other first-round action, seventh-seed Kevin Anderson of South Africa lost to Lukas Lacko of Slovakia 6-4, 7-6 (2), and unseeded Horacio Zeballos of Argentina beat qualifier Ryan Harrison of the United States 6-1, 4-6, 6-4.