Gundogan then added a second on 53 minutes to give Man City a resounding lead in the tie, with the second leg at the Etihad Stadium coming up in three weeks' time.
"The result was maybe perfect but the way we played there are still plenty of things to improve," Gundogan told BT Sport after the final whistle.
"It's Champions League and it's always tough - we didn't concede which was the most important thing after these goals. It seemed comfortable but it was tense and Basel had chances to score."
Man City continue to fight on four fronts as they lead the Premier League table by 16 points and have reached the final of the EFL Cup as well as the fifth round of the FA Cup.