Belgium's Flipkens had little trouble against Shahar Peer of Israel in a dominating 6-2, 6-2 victory, while Swedish veteran Arvidsson dispatched Slovakian qualifier Jana Cepelova 6-4, 6-4 in Memphis.
Also moving was last year's runner-up, New Zealand's Marina Erakovic, who logged a 4-6, 7-6 (8-6), 6-4 decision over Germany's Annika Beck.
Jamie Hampton of the United States advanced as well with a 6-3, 6-3 straight-set victory over Spanish wild card Garbine Muguruza.
American wild card Victoria Duval and Ukrainian Lesia Tsurenko also recorded triumphs Monday.
Celebrity Families of 2014 [PHOTOS]