We present the cosmological analysis of the configuration-space anisotropic clustering in the completed Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey (eBOSS) Data Release 16 galaxy sample. This sample consists of luminous red galaxies (LRGs) spanning the redshift range 0.6< z< 1, at an effective redshift of z(eff = 0.698.) It combines 174 816 eBOSS and 202 642 BOSS LRGs. We extract and model the baryon acoustic oscillation (BAO) and redshift-space distortion (RSD) features from the galaxy two-point correlation function to infer geometrical and dynamical cosmological constraints. The adopted methodology is extensively tested on a set of realistic simulations. The correlations between the inferred parameters from the BAO and full-shape correlation function analyses are estimated. This allows us to derive joint constraints on the three cosmological parameter combinations: D-M(z)/r(d), D-H(z)/r(d), and f sigma(8)(z), where DM is the comoving angular diameter distance, D-H is the Hubble distance, r(d) is the comoving BAO scale, f is the linear growth rate of structure, and sigma(8) is the amplitude of linear matter perturbations. After combining the results with those from the parallel power spectrum analysis of Gil-Marin et al., we obtain the constraints: D-M/r(d) = 17.65 +/- 0.30, D-H/r(d) = 19.77 +/- 0.47, and f sigma(8) = 0.473 +/- 0.044. These measurements are consistent with a flat Lambda cold dark matter model with standard gravity.