The runaway Premier League leaders destroyed Basel's morale with a three-goal burst after just 23 minutes of the first leg last-16 mismatch.
City's first three goals came in a devastating nine-minute spell thanks to a header from Gundogan, a Bernardo Silva volley and a long-range Sergio Aguero strike.
It was Aguero's 14th goal in 11 games in 2018.
Gundogan added a fourth after the break and was only denied a hat-trick by the dogged determination of Basel goalkeeper Tomas Vaclik.
The tie looks all but settled ahead of the return leg at the Etihad on March 7.
Gundogan says City can do better and he threw down the gauntlet to their Champions League rivals.
"The result was maybe perfect, but there are still plenty of things to improve," said Gundogan
The 27-year-old German singled out City's playmaker Kevin de Bruyne for his part in his opening goal.
"I had the opponent on my back so it was not really easy to head it, but the timing was perfect and the ball was perfect from Kevin," said Gundogan.
Pep Guardiola's City starved Basel of the ball, enjoying 69-percent possession.
"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.
"If we get to the quarter-finals that is already better than last season and that is the first step for us."
The Spaniard heaped praise on Gundogan and his midfield team-mates, including David Silva who made a return off the bench after injury.
"Ilkay is a guy with special quality, he is an amazing midfield player who has a sense for goal," said Guardiola.
"He was out for six or seven months last season and we missed him, he is an outstanding player who doesn't feel the pressure.
"With De Bruyne and David Silva, we have a fantastic midfield trio."
This was City's 34th win in all competitions this season -- one more than they managed in the whole of 2016-2017.
City's captain Vincent Kompany made his first Champions League appearance this season after a calf injury. He urged his team to give themselves a rousing victory margin in the second leg.
"If we have this game at the Etihad and it's a 3-2 or a 2-2 draw that won't give us a good feeling to go to the next round," said the burly Belgian.
"We have showed we were the better team today and only for a lack of motivation or effort can we not do the same at home."
Guardiola said the return of David Silva and Leroy Sane, who both came on in the second half, boosts City's chances in the coming weeks.
They play Wigan away on Saturday, face Arsenal in back-to-back matches starting with the League Cup final on February 25, then host Chelsea on March 4.
Sane's return from an ankle injury after only two weeks was a surprise with the German back - a month ahead of schedule.
"Leroy Sane lacks rhythm, but he showed a lot of courage to be here with us," said Guardiola.
Both Kompany and Guardiola criticised Basel's Sankt Jakob Park pitch.
"I thought the pitch was a bit difficult, it was freezing out there," said Kompany while Guardiola described the pitch as "dangerous".
Basel boss Raphael Wicky said the first leg has been effectively decided by City's first-half goal flurry.
"They were very clinical, the first few goals were given away too easy and when you are 3-0 down after 23 minutes, it's obviously over," said the Swiss.