Cluster strong lensing cosmography is a promising probe of the background geometry of the Universe and several studies have emerged thanks to the increased quality of observations using space- and ground-based telescopes. For the first time, we used a sample of five cluster strong lenses to measure the values of cosmological parameters and combine them with those from classical probes. In order to assess the degeneracies and the effectiveness of strong-lensing cosmography in constraining the background geometry of the Universe, we adopted four cosmological scenarios. We found good constraining power on the total matter density of the Universe ({OMEGA}m) and the equation of state of the dark energy parameter w. For a flat wCDM cosmology, we found {OMEGA}m=0.30^+0.09^0.11 and w=-1.12^+0.17^0.32 from strong lensing only. Interestingly, we show that the constraints from the cosmic microwave background (CMB) are improved by factors of 2.5 and 4.0 on {OMEGA}m and w, respectively, when combined with our posterior distributions in this cosmological model. In a scenario where the equation of state of dark energy evolves with redshift, the strong lensing constraints are compatible with a cosmological constant (i.e. {OMEGA}m=-1). In a curved cosmology, our strong lensing analyses can accommodate a large range of values for the curvature of the Universe of {OMEGA}k=0.28^+0.16^0.21. In all cosmological scenarios, we show that our strong lensing constraints are complementary and in good agreement with measurements from the CMB, baryon acoustic oscillations, and Type Ia supernovae. Our results show that cluster strong lensing cosmography is a potentially powerful probe to be included in the cosmological analyses of future surveys.