  We propose a viable dark energy scenario in the presence of cubic Horndeski
interactions and a standard scalar-field kinetic term with two exponential
potentials. We show the existence of new scaling solutions along which the
cubic coupling $G_3$ provides an important contribution to the field density
that scales in the same way as the background fluid density. The solutions
finally exit to the epoch of cosmic acceleration driven by a scalar-field
dominated fixed point arising from the second exponential potential. We clarify
the viable parameter space in which all the theoretically consistent conditions
including those for the absence of ghost and Laplacian instabilities are
satisfied on scaling and scalar-field dominated critical points. In comparison
to Quintessence with the same scalar potential, we find that the cubic coupling
gives rise to some novel features: (i) the allowed model parameter space is
wider in that a steeper potential can drive the cosmic acceleration; (ii) the
dark energy equation of state $w_{\phi}$ today can be closer to $-1$ relative
to Quintessence; (iii) even if the density associated with the cubic coupling
dominates over the standard field density in the scaling era, the former
contribution tends to be suppressed at low redshifts. We also compute
quantities associated with the growth of matter perturbations and weak lensing
potentials under the quasi-static approximation in the sub-horizon limit and
show that the cubic coupling leads to the modified evolution of perturbations
which can be distinguished from Quintessence.
