With the Gunners held to a 2-2 draw at Meadow Park in the first leg of the second-round qualifier, they had to win in Amsterdam to progress.
Miedema fired into the bottom corner in the second half to set Arsenal on their way, although Ajax carried a threat.
The hosts enjoyed a spell of pressure late on, which made for a nervy ending.
It was a slow start to the match, which took place after officials and ground staff had to adjust the height of the goalposts, which were found to be uneven before kick-off.
Jonas Eidevall’s visitors were not as fluent as they have been so far in the Women’s Super League this season and were often sloppy in possession.
Fresh from their victory over rivals Tottenham, Arsenal were hoping to put in a similar performance to keep their European hopes alive.
But it was Ajax who had the first chances of the night when Ashleigh Weerden’s dangerous cross failed to find a team-mate, before Chasity Grant poked wide when through one-on-one with the goalkeeper.
Stina Blackstenius had a couple of chances to test Ajax goalkeeper Lize Kop in the first half, while Miedema fired over the bar.
Ajax, desperately chasing an equaliser in the final 20 minutes, came close when a backheel from Nadine Noordam set up Tiny Hoekstra but she could not provide the finish.
It was a wet night in Amsterdam, where England manager Sarina Wiegman was watching on 24 hours after naming her squad for their upcoming friendlies with the USA and the Czech Republic.
She will have been concerned by the sight of England forward Beth Mead, on as a substitute, going off after a collision to the face.
There was further injury worry for Arsenal as key Brazilian defender Rafaelle Souza was forced off with an ankle injury in the first half.