Levine was looking to get past the second round in Melbourne for the first time and made a promising start, claiming the opening set 6-2 to take the lead.
Simon responded to level the match in the next set, breaking twice in the second with a resolute display at the baseline.
The Frenchman was pushed all the way in a pivotal third set, with Levine rediscovering his form on serve to take it to a tie-breaker.
Both players tightened up under pressure but it was Levine who buckled, dropping crucial points on serve to hand Simon the third set and with it the initiative.
Levine looked dejected and was broken twice in the fourth as Simon finished the job in just under three hours, winning the contest 2-6 6-3 7-6(4) 6-2.