Leading 1-0 from the first leg, Real doubled their advantage in the tie when Luka Modric pounced on Atalanta goalkeeper Marco Sportiello’s poor clearance to square for Karim Benzema to sweep home.
Skipper Sergio Ramos added Real’s second from the penalty spot after Vinicius Junior was cynically tripped just inside the area by Rafael Toloi.
The Italian side pulled one back through Luis Muriel’s marvelous 25-yard free-kick but Marco Asensio’s near-post finish quickly silenced any thoughts of a late fightback.
“The important thing was that we got through to the last eight, which we haven’t done in the last two years,” said midfielder Modric.
“We’ve shown tonight that there’s still a lot of hunger within this team and that we want to continue to win. We’re capable of going really far.”
Madrid, who won the Champions League four times in five seasons between 2014 and 2018, had been eliminated at the last-16 stage the past two years but never looked in danger of succumbing to a similar fate this time around.
Away, specialists, Atalanta had won all four of their Champions League games on their travels this season and could have erased their first-leg deficit within three minutes but Robin Gosens could only divert Muriel’s pull-back straight at Thibaut Courtois from six yards.
Real settled after the let-off but they needed Sportiello’s error to break the deadlock – the keeper had been preferred to Pierluigi Gollini for this return leg but fired a poor ball straight to former Tottenham midfielder Modric, who picked out Benzema to do the rest.
Benzema and Ramos had missed the first leg through injury but Real have not been knocked out of the Champions League with their captain in the side since 2015 and he sealed their progress after Toloi illegally ended Vinicius’ run.
It was the second dazzling dribble in a matter of minutes from the Brazilian striker, who had weaved his way past three defenders into the box shortly earlier only to prod wide with just Sportiello to beat.
Benzema nodded against the post after Sportiello had denied his initial header before an Atalanta improvement that forced two smart Courtois saves from substitute Duvan Zapata was rewarded by Muriel’s superb set-piece strike.
However, Asensio soon settled any potential nerves and while greater tests will await the 13-time European champions in rounds to come, Zinedine Zidane’s side showed they possess the firepower, particularly via the in-form Benzema, to trouble any of the clubs remaining in the competition.
The France forward’s strike moved him on to 70 Champions League goals, fifth in the all-time list, and the ageless attacker has now scored in the competition for 17 consecutive years, matching a mark set by Raul and Lionel Messi.