A simple and robust method to generate a dual-wavelength mode-locked laser using a tunable Mach-Zehnder filter (TMZF) and a single-wall carbon nanotube (SWCNT) based saturable absorber (SA) is proposed and demonstrated. The proposed laser uses a thulium-doped fiber for lasing in the two-micron region and exploits the interferometric spectrum of the TMZF to produce dual peaks with nearly equal magnitude. SWCNT based SA enables mode-locking at a threshold value of 150.4 mW with distinct dual-wavelength peaks at 1919.2 nm and 1963.7 nm. The peaks have a calculated pulse width of 1.8 ps and 1.6 ps, respectively with a repetition rate of 9.1 MHz with a relatively high optical-signal-to-noise ratio value of 59.1 dB. The output is also observed to remain unchanged over time, indicating high stability. The proposed laser has a promising application, particularly in ultrafast gas molecular spectroscopy and sensing.