The dramatic late winner came after Messi had already scored on two free-kicks for the first time in his career, and Sevilla had rallied back from three goals down in the second half.
Ever Banega had given Sevilla a surprise lead with a free-kick of his in the third minute, but Messi quickly reversed the scoreline in Tbilisi.
Messi’s second goal was impressive as he curled the ball in from 30 yards to beat Sevilla goalkeeper Beto into the top corner in the 15th minute.
Barcelona added a third goal just before half-time when Suarez broke away free on goal but had his shot saved by Beto. Suarez then collected the ball and crossed to Rafinha to tap home.
Suarez found the net himself after being played through by Sergio Busquets in the 52nd minute to make it 4-1 but Sevilla were not finished.
Jose Antonio Reyes responded five minutes later and Kevin Gameiro closed the deficit to one with a penalty in the 72nd minute.
Sevilla kept pushing and found the equaliser nine minutes from time through Ciro Immobile.
Pedro’s winner came as Messi was lining up another free-kick, which initially rebounded off the wall and came back to Messi, whose second shot was saved by Beto, only for Pedro to put the ball in the roof of the net.