Tsonga safely negotiated a tricky opener against Phillipp Kohlschreiber on Tuesday to progress.
The French fourth seed, in action on his 27th birthday, celebrated the occasion with an impressive 6-2 6-4 victory.
For the second day running, Spaniard Verdasco rallied from a set down before prevailing 3-6 6-3 6-1 against Croatia's Ivan Dodig.
In first round action, Stanislas Wawrinka defeated 10th seed Feliciano Lopez 6-1 6-4, Andreas Seppi saw off Victor Hanescu 6-3 6-1 to set up a meeting with world number one Novak Djokovic, while 14th-seeded Florian Mayer was upset 3-6 6-1 6-3 by Mikhail Kukushkin.
There were also first round wins for Fabio Fognini, Thomaz Bellucci, Filippo Volandri and Frederico Gil, while Potito Starace progressed after opponent Carlos Berlocq retired hurt while trailing 4-2 in the first set.