In an open letter to Apple , Taylor Swift said she was withholding the record as she was unhappy with the three-month free trial offered to subscribers.
Now Apple says it will pay artists for music streamed during trial periods.
" We hear you @taylorswift13 and indie artists. Love, Apple ," tweeted executive Eddy @Cue .
Swift had said the plan was "unfair", arguing Apple had the money to cover the cost.
"I find it to be shocking, disappointing, and completely unlike this historically progressive and generous company," the 25-year-old said, describing Apple as one of her "best partners in selling music".
"Three months is a long time to go unpaid, and it is unfair to ask anyone to work for nothing.
"We don't ask you for free iPhones. Please don't ask us to provide you with our music for no compensation."
Apple Music launches on 30 June. It will cost $9.99 (£6.30) per month in the US for one person or $14.99 for families.
The singer is due to start the UK leg of her world tour on 23 June
After hearing news of the company's decision on Sunday night Swift tweeted : "I am elated and relieved. Thank you for your words of support today. They listened to us.''
Mr Cue said he did not know if the Shake It Off singer would now make her album available on Apple Music - but he praised her for taking a stand.
"When I woke up this morning and I saw Taylor's note that she had written, it really solidified that we needed to make a change," he told Associated Press.
The music industry has generally supported the company bringing its vast music library to paid streaming, and the firm said it would pay 70% of the music subscription revenue to music owners.
Eddy Cue called Swift in Amsterdam to tell her they had made the change
Speaking to Billboard magazine Cue said they had already been been hearing "a lot of concern from indie artists about not getting paid during the three-month trial period" before Swift spoke out. But he said "we never looked at it as not paying them".
"We had originally negotiated these deals based on paying them a higher royalty rate on an ongoing basis to compensate for this brief time", he said.
They will now pay artists during the trial period and "also keep the royalty rate at the higher rate."
He said he spoke to Swift himself to let her know they were making the changes and "she was thrilled and very thankful".
