Unics held a fourth quarter advantage as Keith Langford led the way with a 23-point haul, only to be denied at the last second by Llull's late jumper.
Real are second in Group A behind Anadolu Efes Istanbul, who had a comparatively easier time in preserving their own 100 per cent record on the road by beating Dinamo Banco di Sardegna 82-75.
The home team tightened up the margin to 57-56 in the third quarter as they made a spirited bid for a maiden Euroleague win before Efes closed out the game in the final quarter.
Experienced centre Nenad Krstic scored 15 points, while Matt Janning and Stephane Lasme added 14 apiece.
The other Group A match saw Zalgiris Kaunas get their campaign up and running in emphatic fashion with a resounding 97-63 victory at home to Nizhny Novgorod.
Unicaja kept pace with Group B leaders CSKA Moscow in dramatic style as Kostas Vasileiadis hit a three-pointer on the buzzer to secure a 87-84 triumph over ALBA Berlin.
Ryan Toolson registered 13 points and eight assists for the victors.
Elswhere in the group, Limoges CSP beat Cedevita Zagreb 71-60 in their first Euroleague home encounter since 1998.
Olympiacos are the only Group D team with two wins from their opening games thanks to a 63-57 result over Laboral Kutxa Vitoria.
Matt Lojeski led the way with 15 points and four steals for the hosts before Bryant Dunston (10 points, six rebounds) proved the difference at both ends.