The Indonesian club was initially ruled to be ineligible for the playoff as they were participating in the unsanctioned Indonesian Super League. But a successful appeal to the Court of Arbitration for Sport saw them reinstated to the Champions League earlier this month.
Centreback Nigel Boogaard scored from a corner kick in the 12th minute, netting the rebound after a header by Evgeniy Levchenko was cleared off the line.
Adelaide continued strongly in the first half with midfielder Iain Ramsey hitting the post with a shot from the left in the 13th minute.
The Indonesian side came close to equalising in the 29th minute when a quick one-two put Titus Bonai free of the last defender but the ball was just out of his reach and he could only produce a weak shot.
Adelaide's Sergio Van Dijk, a Dutch striker of Indonesian descent, came close to scoring a number of times but Persipura goalkeeper Yoo Jae-Hoon was more than up to the mark.
However, the South Korean was beaten again through a corner in the 57th minute when Evgeniy Levchenko volleyed the ball into the net after a flick-on by Van Dijk.
The 29-year-old striker got his name on the scoresheet yet again six minutes before the end as he smashed a left-footed shot past Persipura's goalkeeper in the 84th minute.
Adelaide advance to Group E in the ACL group stage alongside Uzbek side FC Bunyodkor and Japanese heavyweights Gamba Osaka. The fourth team in the group will be the winner of Saturday's play-off between 2009 winners Pohang Steelers of Korea Republic and Thai Premier League runners-up Chonburi.
Japanese side Gamba Osaka are Adelaide's most famous continental rivals as the two played in the Champions League final in 2008 where the Australian team lost 5-0 in aggregate. Later in the year, Gamba Osaka also eliminated Adelaide from the Club World Cup championship.
Adelaide assistant manager Luciano Trani was happy with the win and said that the confidence from the performance could have a positive effect for the match against rivals Sydney FC on Sunday.
"In the end it was (a good performance) but in the first half we had a couple of opportunities to make some of those openings count but unfortunately we either took a wrong option or wasted a bit too much time setting up for a strike on goal," Trani told Footballaustralia.com.
"This game was brought upon us so we needed to physically get ourselves up and mentally be prepared to back up after Gold Coast."
"That's two victories in a row now which we haven't done for a long time, so that will obviously hold us in good stead for Sunday's game against Sydney."