Romanian Mihaela Buzarnescu thrilled her home crowd by rallying in the second set to beat Tamara Zidansek in Bucharest.
Sixth seed and Luxembourgish veteran Gilles Muller saw off Marcos Baghdatis 7-6 (7-5) 6-3 in Newport on Tuesday.
Fernando Verdasco and David Ferrer were victorious in Bastad but Benoit Paire tumbled out in Umag.
Tournament organisers have revealed there will be a five per cent increase for this year's grand slam at Flushing Meadows.
Samantha Stosur looked in impressive form at the Ladies Championship Gstaad, while home favourite Sorana Cirstea triumphed in Bucharest.
Angelique Kerber did not contend in a single grand slam in 2017, but she targeted the Wimbledon title from the off this year.
Following criticism of the lack of a fifth-set tie-break at Wimbledon, the Australian Open has welcomed dialogue with players on the issue.
Although he came up short in the Wimbledon final, Kevin Anderson is confident he can continue to compete at the top of the sport.
Seeded eighth at the Hall of Fame Tennis Championships, Denis Kudla cruised through his opening match.
Leonardo Mayer suffered another first-round exit in Bastad, losing to the 75th-ranked Matteo Berrettini after winning the opening set.