Pep Guardiola's side effectively settled the last-16 tie after just 23 minutes by racing into a 3-0 lead at Basel's Sankt Jakob Park.
Their first three goals came in a devastating nine-minute spell thanks to a Gundogan header, a Bernardo Silva volley and a long-range Sergio Aguero strike.
Basel's misery continued just after the break when Gundogan deftly added a fourth from outside the area and he could easily have finished with a hat-trick.
"To win 4-0 away in the last 16 is an amazing result, we are almost in the quarter-finals, but I am not saying we are already there, out of respect for Basel," said Guardiola.
"We defended well and we were clinical - we had four chances and took them all."
The scoreline gives the Swiss side little hope for the return leg at the Etihad on March 7.
As expected, Basel were starved of the ball as City enjoyed 69 percent possession and completed 754 of their 882 passes.
"For a Champions League game, we did ever so well," said City captain Vincent Kompany after his first Champions League appearance this season.
"In these kinds of situations, it is about more than going through -- if we have a draw at home that won't give us a good feeling."
Germany midfielder Gundogan gave Basel a masterclass in defensive midfield play, complementing his two-goal display by repeatedly snuffing out fledgling Swiss attacks.
The only black mark against him was a late yellow card after a clash with Basel's Ivory Coast veteran midfielder Serey Die.
Basel head coach Raphael Wicky opted to play a back five when the host were defending, but the ploy only failed to keep City out for the first 14 minutes.
Having already had an earlier effort tipped over the bar, Gundogan got in front of Fabian Frei to power home his header when Kevin De Bruyne whipped in City's fourth corner of the game.
The visitors' second goal followed just four minutes later.
Bernardo Silva chested down Raheem Sterling's deflected cross and fired past Basel goalkeeper Tomas Vaclik.