The Promise Keepers were trailing 3-0 from the tie played at the same stadium on Wednesday, before it was abandoned after fans invaded the pitch.
A ruling from the Akwa Ibom FA mandated that the abandoned game should be continued from where it was stopped.
The remaining minutes of the fixture were played on Friday morning, and Algon Boys stood tall to seal their progress to the semi-final. Though they conceded a goal in the process.
It was Paul Obata that reduced the deficit for the Promise Keepers but it was not enough to rescue the two-time Federation Cup champions.
This is the first time in recent seasons that Akwa United will not make it to the final of the state FA Cup and they join Heartland and FC IfeanyiUbah among the high-profile teams that have bowed out of the competition at the state level.
Algon Boys will face Cofine FC in the semi-final in Uyo after the Nationwide League side beat Akwa Starlets 1-0 in the other quarter-final played on Thursday.