Rory Best scored two tries but Ulster, needing a bonus-point victory to keep alive its slim hopes of qualifying for the knockout stages, could only manage a 24-24 draw with Munster at Thomond Park.
That result made sure Edinburgh was certain to progress before its game with Glasgow Warriors kicked off at Murrayfield.
Still, James Johnstone, Jordan Lay and Duhan van der Merwe all touched down in the first half to make sure Richard Cockerill's side secured the 1872 Cup with a 24-19 win on home soil.
It will next travel to play Munster, which finished in second place in Conference A, next weekend.
Scarlets secured a home tie in the opening round of the play-offs thanks to a 33-8 result against the Dragons at Principality Stadium.
The defending champion scored five tries, four of them coming after half-time, to secure a bonus-point victory against its Welsh rival.
Its reward is a clash with the Cheetahs, who recorded a 29-20 result against fellow South African franchise Southern Kings in Port Elizabeth, in the next round.
Dan Biggar was the hero for Ospreys in its dramatic 26-23 victory over Cardiff Blues, with the five-eighth landing a late drop-goal to settle the all-Welsh clash.
Biggar's late three-pointer took his personal tally to 16 points, including a 54th-minute try that helped his side overturn a seven-point half-time deficit.
Connacht marked retiring captain John Muldoon's 327th and final game with a resounding 47-10 triumph over Leinster.
Muldoon, who is hanging up his boots to join Bristol as its new defence coach, kicked a late conversion as the host crushed the Champions Cup finalist, which fielded a much-changed side knowing it was already certain to top Conference B.
Zebre, meanwhile, finished their campaign with a third successive win, seeing off Benetton Treviso 22-17 in the Italian derby at Stadio Monigo.