The England international thought he had a hat-trick at the death at the Etihad Stadium, but a review showed Sergio Aguero was offside in the build-up as Tottenham were beaten 4-3 on the night and progressed 4-4 via the away goals rule.
Spurs had already been assisted by VAR when Fernando Llorente bundled in from close range 17 minutes from time, which ultimately booked a semi-final date with Ajax.
A chaotic first half had City 3-2 up at the break, as Sterling wiped out Son Heung-min's brace then scored the hosts' third following a fortunate Bernardo Silva equaliser.
Aguero's brilliant strike had seemingly put City into the semis before VAR deemed Llorente's apparent handball to be accidental and Pep Guardiola's men saw wild celebrations for Sterling's last-gasp effort curtailed in sensational scenes.
A rip-roaring start astonishingly produced four goals in the first 11 minutes, with City initially overturning Spurs' first-leg lead through Sterling's gorgeous curling finish just inside the area.
