Real forged ahead with a goal of genuine quality as a superb long ball forward was flicked back by Mesut Ozil for Karim Benzema to fire high into the net.
And Jose Mourinho's men doubled their lead before the break, with former Liverpool defender Alvaro Arbeloa driving home from range via the aid of a slight deflection.
Cristiano Ronaldo added further gloss to the scoreline, firing home after some miserable play from Ajax gifted possession to the visitors before hitting a fourth from the spot.
The only note of concern for Mourinho was a pair of late red cards, with Xabi Alonso and Sergio Ramos dismissed for second bookable offences in the closing stages.
Arsenal were unable to shake off the malaise of their North London derby reverse to Spurs as they suffered a second successive Group H defeat, with Braga now level on points with the Gunners.
The Gunners succumbed to two late goals from Matheus in Portugal.
The Brazilian striker broke the offside trap before firing beyond Lukasz Fabianski in the 83rd minute before producing a splendid finish following a driving run in the third minute of added time.
Arsene Wenger's side must beat FK Partizan, who have lost all five of their matches to date, at Emirates Stadium on 8th December to be sure of booking their place in the knockout phase of the competition.
Shakhtar Donetsk are the new leaders of Group H and have 12 points on the board following their 3-0 success away to Partizan.
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