The victory was Madrid's fifth in a row in all competitions and their fourth straight league success. As a result, they will have the chance to move above Atletico when the pair lock horns next weekend.
Alaves arrived at the Bernabeu having lost just seven of their 21 league matches this season, with Abelardo's side starting the weekend in fifth but dropping to seventh ahead of kickoff.
It has been a remarkable campaign for the Basque outfit thus far, and they did record a 1-0 victory over the European champions in the reverse match earlier this season, but they could not make it a league double on Sunday night.
Dani Ceballos had Real's first chance of the match in the third minute, but the Spaniard saw his goal-bound effort deflected over the top of the Alaves crossbar.
Ceballos then sent one high of the target in the fifth minute before Benzema bent one wide of the post three minutes later as the home side started full of confidence at the Bernabeu.
Vinicius has been outstanding for Los Blancos in recent weeks and was next to come close with a powerful strike in the 17th minute, but Alaves goalkeeper Fernando Pacheco was on hand to make a smart save.
Madrid did make the breakthrough in the 30th minute, however, with Benzema turning a low Reguilon cross into the back of the net to continue his wonderful form in front of goal.
It was no more than the home side deserved, having dominated the first half against an Alaves team who were struggling to support lone forward Jonathan Calleri in the final third of the field.
Both teams had opportunities in the early stages of the second period, but Ruben Duarte's volley just missed the post before Pacheco kept out a half-volley from Benzema down the other end of the field.
Pacheco then had to deny Gareth Bale in the 50th minute, with the Welshman having a rare chance to release a strike towards the bottom corner of the net.
Alaves had a brilliant chance to level in the 53rd minute when the ball fell kindly for Burgui in a dangerous position, but the attacker could only fire the ball past the post, before Calleri fired over four minutes later.
Bale sent a free kick just over the Alaves crossbar in the 63rd minute, and that proved to be Welshman's final piece of action as Marco Asensio replaced him.
Alaves had a brilliant chance to level in the 66th minute when Jony broke into the Madrid box, but Thibaut Courtois was on hand to make an excellent save. Vinicius then sent one over the Alaves crossbar 20 minutes from time as the two teams continued to be separated by just the one goal.
However, Vinicius doubled Madrid's lead in the 80th minute when the Brazilian collected a cross from Asensio before expertly sliding the ball into the bottom corner of the net to put the result beyond doubt.
Real were not finished, however, and after Alvaro Odriozola had a goal chalked off for offside, the full-back provided a wonderful cross for Mariano to head a third in the first minute of added time at the end of the 90.
Madrid's next three matches are potentially season-defining. Indeed, they will travel to Barcelona for the first leg of their Copa del Rey semi-final next week, before visiting Atletico in the league next weekend. Solari's side will then travel to Ajax for the first leg of their Champions League last-16 clash on February 13.
REAL MADRID (4-3-3): Courtois; Odriozola, Nacho, Ramos, Reguilon; Ceballos, Casemiro, Modric; Bale (Asensio 63'), Benzema (Mariano 75'), Vinicius (Isco 87')
ALAVES (4-1-4-1): Pacheco; Duarte, Maripan, Laguardia, Vigaray; Pina; Jony, Wakaso, Manu (Guidetti 81'), Burgui (Rolan 68'); Calleri (Blanco 82')body check tags ::