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The Italian side managed to get one goal back thanks to a free kick from Luis Muriel in the 83rd but any hope they had of making a comeback was quickly dashed a minute later by a low strike from Real substitute Marco Asensio.
Atalanta had pulled off away wins over European giants Liverpool and Ajax in the group stage and nearly got off to an ideal start when Robin Gosens met a Muriel cross in the third minute but was thwarted by goalkeeper Thibaut Courtois.
The Serie A side continued to harass Real with their high pressing but the hosts slowly managed to pass their way out of trouble thanks to the experienced heads of midfield duo Toni Kroos and Luka Modric.
Real found the breakthrough on the night thanks to some high pressing of their own which saw Modric stick out a leg to block a kick out from Atalanta keeper Marco Sportiello and then square the ball to Benzema, who calmly stroked it into the net.
The home side's Brazilian forward Vinicius Jr should have doubled their lead after a superb move involving left back Ferland Mendy but after doing the hard part to coast past Atalanta's defence he missed the target from close range.
He made amends by bursting into the area and drawing a foul in the box by Rafael Toloi, allowing Ramos to blast just beyond the outstretched Sportiello and effectively end the tie as a contest, scoring his 101st goal for the club in the process.