Berlocq, who won the Portugal Open this month, broke for a 5-3 lead in the final set to overcome Julien Benneteau of France 6-1, 4-6, 6-3, while Klizan rallied past Bernard Tomic of Australia 4-6, 6-3, 6-4.
Tomic slammed 11 aces but dropped serve in the final game against Klizan, who won the BMW Open this month in Germany.
Robin Haase of the Netherlands, Dominic Thiem of Austria and American qualifier Jack Sock also advanced.
Haase did not face a break point in his 6-3, 6-2 win over Filippo Volandri of Italy, Thiem outlasted Steve Johnson of the United States 3-6, 7-6 (5), 7-5, and Sock saved all five break points he faced in his 6-4, 6-2 win over Kenny De Schepper of France.
Third-seeded Gael Monfils of France pulled out from the singles' main draw after feeling pain in his right ankle in a doubles match. Monfils and compatriot Benoit Paire lost 6-4, 6-7 (3), 10-2 to Daniele Bracciali of Italy and Oliver Marach of Austria on Tuesday. The facetious Monfils shot and posted a video of himself and Paire on Instagram during a changeover. Sam Querrey of the United States replaced Monfils in the main draw as a lucky loser.