Real pushed forward through Vinicius Junior, whose shot on the edge of the area drew a fine save from Alaves shot-stopper Fernando Pacheco on 18 minutes.
With 30 minutes gone, the breakthrough arrived courtesy of a simple tap-in from Benzema following Sergio Reguilon’s square pass.
Los Blancos held on for the lead well into the break, with little threats coming through from the opposition.
Renewed optimism for the second half coincided with Alaves' first opening inside the second half, as Ruben Duarte shaved the post.
After 54 minutes, a loose ball fell to Burgui in the Real penalty area but the attacker failed to beat Courtois from a promising situation.
Burgui posed another threat on 65 minutes with a blistering left-footed strike, but enterprising reflexes from Courtois saved the day for the European champions.
The home side eventually doubled their on 80 minutes as Vinicius collected a deep ball from Asensio to tuck home for his second league goal of the season.
Second-half substitute Mariano Diaz sealed the match two minutes into added time with an easy finish at the back post following a cross-field delivery.
With Atletico Madrid going down to Real Betis earlier in the day, Real have now closed the gap on the second-place side to two points.