PSV made a remarkable comeback over Heerenveen, trailing 2-3, they scored two in the final two minutes to claim victory.
Lasse Shoene made up for his missed PK by scoring the crucial winner late in the second half to keep Ajax in the title hunt.
Odegaard, who joined Madrid at the age of 16 in January 2015, has made only one substitute La Liga appearance.
Ross Dyer delves into all the latest from American footballers abroad, featuring Tijuana's Paul Arriola and Joe Corona.
Justin Kluivert, the son of former Netherlands international Patrick Kluivert, made his Ajax debut during a 3-1 Eredivisie win against PEC Zwolle.
Narsingh, who has signed a two-and-half-year deal at the Liberty Stadium, said he believed the Swans could get clear of relegation danger.