World No. 30 Carreno Busta won a tense first set by breaking Italian Sinner’s serve at 5-5 before serving out the frame.
The second set also saw just one break, this time from Sinner at 4-3. The world No. 79 then successfully held serve to set up a decider.
Carreno Busta, 28, struck early in the third set and established a 3-0 lead, but Sinner broke back after saving three match points as the Spaniard tried to serve out the contest at 5-4.
The set ultimately progressed to a back-and-forth tiebreak, which Carreno Busta won 8-6 after saving two match points himself.
World No. 20 Auger-Aliassime took the first set by breaking Bedene at 5-4, then edged a dramatic tiebreak in the second. He faces Carreno Busta in the final four on Saturday.
Later on Friday, defending champion Gael Monfils — who took home the trophy in Montpellier earlier in February — comes up against British No. 1 Dan Evans, and Russia’s Andrey Rublev takes on Filip Krajinovic of Serbia.