City entered the match with a 4-0 advantage from the first leg in Switzerland, and triumphed 5-2 on aggregate to ease into the final eight.
City boss Pep Guardiola made six changes to the team that started against Chelsea in the Premier League on Sunday afternoon. Claudio Bravo was handed a spot between the sticks, while Danilo and John Stones came into the back four.
It took City just eight minutes to make the breakthrough at the Etihad Stadium, with Gabriel Jesus tapping a low Bernardo Silva cross into the back of the net from close range for his first goal since November.
The visitors were causing problems in the final third, and they deservedly levelled in the 17th minute when Mohamed Elyounoussi smashed the ball home from close range after a fortunate deflection inside the City box.
Elyounoussi then fired wide of the post down the other end as both teams continued to have joy in the final third.
City had 70% of the ball in the first period, but Basel more than deserved to enter the half-time break on level terms following an impressive first-half performance in Manchester.
The first 15 minutes of the second period passed without incident as both teams continued to concede possession in forward areas.
The home side's supporters were becoming restless as the match headed for its final 25 minutes with City not too interested in searching for a second goal.
Instead, it was Basel that took the lead in the 72nd minute when Elyounoussi worked space in a wide area before feeding Michael Lang, who found the top corner with a brilliant strike from a narrow angle.
City came close to equalising 10 minutes from time when Brahim Diaz collected a super pass from Sane before striking towards goal, but Vaclik was on hand to make the save.
The hosts had a lot of the ball in the final exchanges, but it was comfortable for Basel as the Swiss outfit picked up a morale-boosting win on their way to exiting the competition.
MAN CITY (4-2-3-1): Bravo; Danilo, Stones, Laporte, Zinchenko; Gundogan (Diaz 66'), Toure, Foden (Adarabioyo 89'); B.Silva, Jesus, Sane
BASEL (5-2-3): Vaclik; Lang, Lacroix, Frei, Suchy, Riveros; Serey Die, Zuffi; Bua (Stocker 68'), Elyounoussi, Oberlin (Van Wolfswinkel 74')