Murray trailed 3-1 in the final set, then held serve and soon broke back to even the score at 3-3 at a covered Ariake Colosseum. With rain falling since morning, play on the outer courts was suspended.
"He played well under the conditions. It took me longer to find the range," Murray said. "The hardest thing for me always is changing from playing outdoors to indoors."
Murray, who won last week's Thailand Open, struggled in the second set as the unseeded Baghdatis used a strong backhand and some clever drop shots to take the upper hand after losing the opening set.
"It was very tough. It was very long," Murray said. "I did a lot of running in the first two sets."
Playing primarily from the baseline, the Briton converted at crucial points down the stretch as Baghdatis appeared to tire and committed several unforced errors.
Baghdatis and Murray played earlier this year in Rotterdam, and the Cypriot won in straight sets on a hard court. Baghdatis still holds a 3-2 edge in five matches against Murray.
"He is a tough guy to play against," Murray said. "He has a big serve. He can hurt you in many ways on the court."
Earlier, Dmitry Tursunov of Russia beat sixth-seeded Janko Tipsarevic of Serbia 7-6 (6), 6-7 (3), 7-5. In the second round, fourth-seeded Mardy Fish of the United States beat Ernests Gulbis of Latvia 6-2, 6-4.