Anderlecht general manager Herman van Holsbeeck had admitted the chances of his side overturning their 3-0 first-leg deficit were "almost nil".
And he was proved absolutely correct as Miralem Sulejmani scored twice in the space of five minutes in the first half to kill the tie.
The visitors actually had the first chance after six minutes when Rocha Kanu fired over the bar from the left side of the penalty area.
Four minutes later Kanu was in the thick of the action again but this time his 20-yard effort was held by Ajax goalkeeper Maarten Stekelenburg.
And 50 seconds later the home side had swept down the other end to open the scoring as Sulejmani drilled home a left-footed shot into the bottom corner from the edge of the area after collecting Siem De Jong's pass.
That understandably deflated Anderlecht and it was no surprise when Sulejmani collected his second of the night five minutes later.
It was another unerring finish from the striker, this time from 12 yards in front of goal after he latched onto a pass from Mounir El Hamdaoui.
Anderlecht could only hope to limit the damage now but almost fell further behind shortly after when Lorenzo Ebecilio, who had been booked after just three minutes, fired over from close range.
Ebecilio did better with his next attempt on 20 minutes but saw his 12-yard shot blocked by Anderlecht defender Roland Juhasz.
The visitors managed to weather the storm and threatened to pull a goal back in the 33rd minute, but Matias Suarez headed wide from six yards from Jan Lecjaks' cross.
To their credit Anderlecht continued to try to find a way back into the game and Romelu Lukaku had the first chance of the second half when his shot was blocked by Gregory van der Wiel.
However, Ajax were not simply content to sit back and protect their lead and Ebecilio tested Silvio Proto in the Anderlecht goal with an 18-yard drive.
Chances eventually became more scarce as the clock wound down and Ajax could celebrate a comprehensive win in front of a crowd of 42,591.
Ajax 2-0 Anderlecht: Match Report - view commentary, squad, and statictics of the game as it happened.