The Liverpool left-back captained his country and played the full 120 minutes of the play-off in Belgrade after Luka Jovic’s last-gasp equaliser forced extra-time following Ryan Christie’s earlier opener.
Nothing could separate the sides in the additional half-hour and a shootout was required, with Scotland winning 5-4 as David Marshall kept out Aleksandar Mitrovic’s kick.
Robertson and co will therefore join England, Croatia and Czech Republic in Group D at Euro 2020, which is scheduled to begin on June 11, 2021.
The result meant disappointment for Reds midfielder Marko Grujic, who was an unused substitute for Serbia.