Peter Bosz's young side will feel aggrieved they only had two goals, both scored by captain Davy Klaassen, to show for their excellent performance.
Ajax threatened as early as the fourth minute when Nick Viergever's acrobatic volley flew high and wide of the Schalke goal.
Ralf Fahrmann was first called into action into the 12th minute by a well-struck low volley from outside the box by Donny van de Beek, which the keeper gratefully gathered at the second attempt.
Eleven minutes later the hosts took the lead. The left-hand channel was a productive area and Amin Younes' run into the box was halted illegally by Alessandro Schopf.
Klaassen stepped up to take the penalty and drilled it into the top corner, giving Fahrmann no chance.
They might have doubled their lead just after the half-hour mark. Davinson Sanchez was agonisingly close to turning in a cross by Klaassen and then, when the ball was sent back in, Sanchez and Nabil Bentaleb both challenged for it.
It was the defender who got the final touch and it needed a fingertip save from Fahrmann onto the bar to prevent an own goal.
Schalke did not manage their first attempt on target until the final moments of the half but Benedikt Howedes' shot was comfortably saved by Andre Onana.
Read the full article on eurosport.co.uk: Klaassen drives classy Ajax towards semis