The spectral evolution and spectral lag behavior of 92 bright pulses from 84 gamma-ray bursts observed by the Fermi Gamma-ray Burst Monitor (GBM) telescope are studied. These pulses can be classified into hard-to-soft pulses (H2S; 64/92), H2S-dominated-tracking pulses (21/92), and other tracking pulses (7/92). We focus on the relationship between spectral evolution and spectral lags of H2S and H2S-dominated-tracking pulses. The main trend of spectral evolution (lag behavior) is estimated with log E-p alpha k(E) log(t + t(0)) ((tau) over cap alpha k((tau) over cap) log E), where E-p is the peak photon energy in the radiation spectrum, t + t(0) is the observer time relative to the beginning of pulse - t(0), and (tau) over cap is the spectral lag of photons with energy E with respect to the energy band 8-25 keV For H2S and H2S-dominated-tracking pulses, a weak correlation between k((tau) over cap)/W and k(E) is found, where W is the pulse width. We also study the spectral lag behavior with peak time t(pE) of pulses for 30 well-shaped pulses and estimate the main trend of the spectral lag behavior with log t(pE) alpha k(tp) log E. It is found that k(tp) is correlated with k(E). We perform simulations under a phenomenological model of spectral evolution, and find that these correlations are reproduced. We then conclude that spectral lags are closely related to spectral evolution within the pulse. The most natural explanation of these observations is that the emission is from the electrons in the same fluid unit at an emission site moving away from the central engine, as expected in the models invoking magnetic dissipation in a moderately high-sigma outflow.