Nishikori was just two points from defeat while trailing 5-2 in the second set, but reeled off five straight games to even the sets before cruising to victory in the final set.
Del Potro, ranked seventh in the world, prevailed with solid play from the baseline and capitalized on some crucial mistakes by Baghdatis.
Baghdatis, ranked 49th, broke serve in the opening game of the match and held throughout to take the first set. The Cypriot mixed it up, frequently coming to the net and using drop shots to try and keep del Potro off balance.
The players exchanged breaks early in the second set, then stayed on serve until the final game, when del Potro broke with a volley to level the sets.
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 here, 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.
Third-seeded Milos Raonic of Canada, who lost to Nishikori in the final last year, will play his first-round match against Go Soeda on Wednesday.
