Davy Klaassen's double earned Ajax a richly-deserved 2-0 victory over Schalke, who were thankful to goalkeeper Ralf Fahrmann for keeping them in the Europa League quarter-final tie.
Ajax have not reached a major European semi-final since 1996-97 Champions League, but Thursday's impressive first-leg performance at the Amsterdam ArenA put Peter Bosz's side on track to be included in the last-four draw.
The only concern for Bosz from this result will be that his side failed to come away with a more comprehensive lead, Fahrmann in inspired form throughout to continually deny the likes of Bertrand Traore and Amin Younes.
Fahrmann could do little about Klaassen's brace, however - the first a thunderous penalty following Alessandro Schopf's foul on Younes in the opening period.
If Klaassen's opener was a triumph for brute force, his second - a superbly taken half-volley soon after the interval - was testament to the captain's finesse.
The pick of Fahrmann's remarkable highlight reel arrived 13 minutes from time with a fingertip stop onto the crossbar from Donny van de Beek - a save that could prove crucial come next week's return in Gelsenkirchen.
Roared on by a vociferous home crowd, Ajax made the brighter start that reaped its rewards midway through the half thanks to Klaassen's emphatic spot-kick.
Younes picked the ball up on the left and darted between Thilo Kehrer and Schopf, who was guilty of a clumsy challenge and Klaassen hammered home into the top-right corner.
It was almost two just after the half-hour as Davinson Sanchez's deflected header rebounded off the top of the crossbar.
Ajax took little time to regain their stride after the interval and doubled their lead in the 52nd minute, Klaassen expertly steering home on the half-volley after Justin Kluivert's cross picked him out unmarked inside the box.
Schalke were rocking, another poor pass out of defence allowing Younes to play in Traore who was superbly denied by the onrushing Fahrmann.
Four minutes later Younes and Traore combined again, but once more Fahrmann made himself big to deny the Chelsea loanee.
Schalke were unable to quell the relentless wave of Ajax attacks, Fahrmann again haring from his line to thwart Younes, before getting down low to keep out Traore's drive.
The visitors showed some signs of life as the game entered its final 20 minutes, Daniel Caligiuri testing Andre Onana with a low drive.
Fahrmann kept out Younes at his near post as Ajax went in search of a third, which almost arrived from the resulting corner, the keeper somehow tipping Van de Beek's rasping drive onto the bar.
Nick Viergever became the latest Ajax man to be denied by Fahrmann two minutes later, the home side forced to settle for a significant but not decisive advantage heading into the second leg.