Single crystals of EuPtSi$_3$, which crystallize in the BaNiSn$_3$-type crystal structure, have been grown by high temperature solution growth method using molten Sn as the solvent. EuPtSi$_3$ which lacks the inversion symmetry and has only one Eu site in the unit cell is found to be an antiferromagnet with two successive magnetic transitions at $T_{\rm N1}$ = 17 K and $T_{\rm N2}$ = 16 K, as inferred from magnetic susceptibility, heat capacity and $^{151}$Eu M\"ossbauer measurements. The isothermal magnetization data for $H \parallel$ [001] reveal a metamagnetic transition at a critical field $H_{\rm c}$ = 1 T. The magnetization saturates to a moment value of 6.43 $\mu_{\rm B}$/Eu above 5.9 T (9.2 T) for $H \parallel $ [001] ([100]), indicating that these fields are spin-flip fields for the divalent Eu moments along the two axes. The origin of this anisotropic behaviour is discussed. A magnetic (H, T) phase diagram has been constructed from the temperature dependence of isothermal magnetization data. The reduced jump in the heat capacity at $T_{\rm N1}$ indicates a transition to an incommensurate, amplitude modulated antiferromagnetic structure. The shape of the hyperfine field split M\"ossbauer spectrum at $T_{\rm N1}$ provides additional support for the proposed nature of this magnetic transition.
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