We perform a joint BAO and RSD analysis using the eBOSS DR16 LRG and ELG samples in the redshift range of z is an element of [0.6, 1.1], and detect an RSD signal from the cross-power spectrum at a similar to 4 sigma confidence level, i.e., f sigma(8) = 0.317 +/- 0.080 at z(eff) = 0.77. Based on the chained power spectrum, which is a new development in this work to mitigate the angular systematics, we measure the BAO distances and growth rate simultaneously at two effective redshifts, namely, D-M/r(d)(z = 0.70) = 17.96 +/- 0.51,D-H/r(d)(z = 0.70) = 21.22 +/- 1.20,f sigma(8)(z = 0.70) = 0.43 +/- 0.05, and D-M/r(d)(z = 0.845) = 18.90 +/- 0.78,D-H/r(d)(z = 0.845) = 20.91 +/- 2.86,f sigma(8)(z = 0.845) = 0.30 +/- 0.08. Combined with BAO measurements including those from the eBOSS DR16 QSO and Lyman-alpha sample, our measurement has raised the significance level of a non-zero Omega(Lambda) to similar to 11 sigma. The data product of this work is publicly available at https://github.com/icosmology/eBOSS_DR16_LRGxELG and https://www.sdss.org/science/final-bao-and-rsd-measurements/.