The cross section for the process $e^+e^-\to \pi^+\pi^-J/\psi$ is measured precisely at center-of-mass energies from 3.77 to 4.60~GeV using 9~fb$^{-1}$ of data collected with the BESIII detector operating at the BEPCII storage ring. Two resonant structures are observed in a fit to the cross section. The first resonance has a mass of $(4222.0\pm 3.1\pm 1.4)$~MeV/$c^2$ and a width of $(44.1\pm 4.3\pm 2.0)$~MeV, while the second one has a mass of $(4320.0\pm 10.4 \pm 7.0)$~MeV/$c^2$ and a width of $(101.4^{+25.3}_{-19.7}\pm 10.2)$~MeV, where the first errors are statistical and second ones are systematic. The first resonance agrees with the $Y(4260)$ resonance reported by previous experiments. The precision of its resonant parameters is improved significantly. The second resonance is observed in $e^+e^-\to \pi^+\pi^-J/\psi$ for the first time. The statistical significance of this resonance is estimated to be larger than $7.6\sigma$. The mass and width of the second resonance agree with the $Y(4360)$ resonance reported by the $BABAR$ and Belle experiments within errors. Finally, the $Y(4008)$ resonance previously observed by the Belle experiment is not confirmed in the description of the BESIII data.