Taylor, a Commonwealth Games silver medallist in Delhi, needed to be the sole British fighter to make the semi-finals in Turkey to guarantee his place and progressed to the final four with a 17-9 win over Armenia's Vladimir Sarukhanyan.
He then watched British champion Maxwell, one of the sport's rising stars, lose 17-8 to home favourite, world number five and European champion Fatih Keles.
Taylor becomes the sixth British boxer to qualify for the Games and will be joined by light heavyweight Callum Smith if he wins one more bout.
Smith beat Hungary's Imre Szello 16-3 to reach the semi-finals, where a victory over Vatan Huseynli would secure the qualifying standard.
And even if he loses he can still make the Games if Huseynli went on to win the final.