The win, a sixth in seven games on home soil this season for the Drua, secured seventh place in the final standings and extinguished the playoff hopes of the reigning champion Canterbury Crusaders and Perth-based Western Force.
They will have to travel outside of their island home in the playoffs, however, with a trip to New Zealand or Australia to play the Auckland Blues, Wellington Hurricanes or ACT Brumbies on the cards next weekend.
The Rebels, whose demise at the end of their 14th season was confirmed on Thursday, will have at least one more Super Rugby match against another of the leading trio after finishing eighth in the final regular season standings.
The Australians were right in Saturday's contest at 19-19 at the break but were kept scoreless in the second half as a three-try blitz within 13 minutes of the restart gave the Drua a comfortable cushion.
Roared on by another big crowd at Churchill Park, Selestino Ravutaumada crossed from the first move of the half, his fellow winger Waqa Nalaga scored his second try seven minutes later and scrumhalf Frank Lomani added the third in the 53rd minute.
"It was a do-or-die game and the boys gave their all," said Drua captain Meli Derenalagi.
"It's a second year in the quarters for us but we talked about wanting to go further, and our preparations for next week start now."
The Crusaders, Super Rugby champions for the last seven seasons, had needed a victory in their final match combined with a Drua defeat to avoid missing out on the playoffs for the first time since 2015.
They had put the first part of the jigsaw puzzle in place on Friday night with a comfortable 43-10 bonus point win over the Moana Pasifika in Christchurch, which ended the playoff hopes of their opponents.
The Queensland Reds, who were already assured of fifth place and a quarter-final against the Waikato Chiefs, finished their regular season on Friday in Sydney with a narrow 27-26 victory over the New South Wales Waratahs, who finished bottom.
The Blues host the Chiefs, the Hurricanes the Highlanders and the Brumbies visit the Force in matches later on Saturday that will decide the remaining quarter-final fixtures.
