Bernard posts three-set win
The match began on Monday but was suspended overnight due to rain with the Australian leading one set to love and 4-3.
When play resumed, the American won four out of five games to take the second set 7-5 and level the match.
In the final set, the teenager held his nerve to secure a break late in the set, which he went on to win 7-5 and seal the win.
Tomic next plays No.8 seed Edouard Roger-Vasselin from France.
The pair met in the final round of Wimbledon qualifying last year with Roger-Vasselin winning in straight sets.