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The Manchester City-bound 22 year-old had effectively secured the win for River in its final game in Group F, scoring a brace in the opening 20 minutes, before sealing a first-half hat-trick four minutes from the interval.
His third was the pick of the bunch, too, retreating between the lines to receive the ball before driving low and hard into the bottom corner from just outside the penalty area.
After Santiago Simon's goal in the 53rd minute, Enzo Gomez arguably topped them all for degree of difficulty, sending the ball into the top corner in the 79th minute before Alvarez rounded out the scoring four minutes later.
Los Millonarios ended the group stage undefeated, claiming 16 points from a possible 18 with the win.
Fortaleza snatches second from Colo Colo
In the other Group F game, Fortaleza took the three points and second spot from Santiago, defeating Colo Colo 4-3.
With both teams even on seven points coming into the final matchday, Los Albos needed the win but started the match in an inadequate fashion, conceding in only the second minute to Silvio Romero.
A Moises double before Yago Pikachu's goal in the 61st minute put Fortaleza in control, despite a brace of own-goals from Brayan Ceballos.
Cerro Porteno scrapes through
Olimpia was unable to leapfrog Cerro Porteno in Group G despite claiming a 1-0 win in Asuncion, putting the latter through to the last 16 on goal difference.
El Decano needed to win by two goals or more in this Libertadores edition of the Paraguayan Clasico, trailing the Clausura champions by three points and with an inferior goal difference of one.
Despite an early goal from Fernando Cardozo in the 10th minute, they were unable to break the Azulgrana down and will have to continue in the Copa Sudamericana, finishing third.