Pe’er, ranked No. 120 in the world, trailed 4-2 in the third and decisive set against Yuliya Beygelzimer.
Pe’er, ranked No. 124 in the world, came back from behind to beat Kristyna Pliskova (129) of the Czech Republic 1-6, 7-6(4), 7-5 in the final of the $50,000 event.
After falling to Croatia in its opener, the blue-and-white suffered its second straight 3-0 thrashing on Thursday, this time at the hands of Belgium.
Israel will face Croatia, Belgium and Latvia this week and will have to top the pool and overcome the winner of Pool A to progress to the World Group II playoffs in April.
Following Pe’er and Glushko’s exits, Dudi Sela (105) will be Israel’s lone representative in the singles events in Melbourne.
Pe’er, ranked No. 117 in the world, beat Canadian Francoise Abanda (202) 3-6, 7-5, 6-3 to set up a meeting on Friday with Anastasia Rodionova (190) of Australia.
Glushko next meets China's Yafan Wang, while Pe'er is set to face Anastasia Rodionova of Australia.
All three of Israel's representatives in the singles tournaments suffer first-round exits.
The 27-year-old Israeli, ranked No. 86 in the world, suffered a 6-3, 3-6, 6-2 defeat to Shuai Zhang (36) of China.
Israeli tennis players were both sent packing in the first round of French Open on Monday.