Jaziri had broken early in the first set to lead 2-0 but Granollers broke back. The Tunisian saved two set points at 5-6 to force a tie-break but he lost it easily 7-1.
The 28-year-old stepped things up in the second set to break for 4-2 and leveled things with Granollers, but the world No23 took advantage of a poor third set from Jaziri to take the lead.
Some sensational hitting from Jaziri saw him capture the fourth set to force a decider, which will be played today - second match on Court 3, not before 12:30 Paris time.
The winner of this match will face world No261 Paul-Henri Mathieu next after the former 12th-ranked Frenchman stunned world No11 John Isner in a marathon second round, that lasted 5 hours 41 minutes and ended 18-16 in the fifth set.