But Viola, having finished eighth in Serie A, had not reached a European final since losing the 1990 UEFA Cup showpiece against Juventus.
His solitary piece of European silverware came in the Cup Winners’ Cup in 1961.
Now they have a chance to end that drought.
González equalized for Fiorentina in the 35th minute when he went unmarked at Cristiano Biraghi’s corner.
Zeki Amdouni equalized for Basel in the 55th minute, volleying past Pietro Terraciano after Andy Pelmard’s free-kick fired past the Swiss forward.
Gonzalez forced extra-time after 72 minutes with a clinical finish into the far corner.
With the penalty only seconds away, Barak latched onto Luka Jovic’s header and slotted the winner.