Graphene supported on a transition metal dichalcogenide substrate offers a novel platform to study the spin transport in graphene in presence of a substrate induced spin-orbit coupling, while preserving its intrinsic charge transport properties. We report the first non-local spin transport measurements in graphene completely supported on a 3.5 nm thick tungsten disulfide (WS$_2$) substrate, and encapsulated from the top with a 8 nm thick hexagonal boron nitride layer. For graphene, having mobility up to 16,000 cm$^2$V$^{-1}$s$^{-1}$, we measure almost constant spin-signals both in electron and hole-doped regimes, independent of the conducting state of the underlying WS$_2$ substrate, which rules out the role of spin-absorption by WS$_2$. The spin-relaxation time $tau_{text{s}}$ for the electrons in graphene-on-WS$_2$ is drastically reduced down to~10 ps than $tau_{text{s}}$ ~ 800 ps in graphene-on-SiO$_2$ on the same chip. The strong suppression of $tau_{text{s}}$ along with a detectable weak anti-localization signature in the quantum magneto-resistance measurements is a clear effect of the WS$_2$ induced spin-orbit coupling (SOC) in graphene. Via the top-gate voltage application in the encapsulated region, we modulate the electric field by 1 V/nm, changing $tau_{text{s}}$ almost by a factor of four which suggests the electric-field control of the in-plane Rashba SOC. Further, via carrier-density dependence of $tau_{text{s}}$ we also identify the fingerprints of the Dyakonov-Perel type mechanism in the hole-doped regime at the graphene-WS$_2$ interface.
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