Tuchel returned to the Bundesliga to take over at Bayern after they made the shock decision to sack Julian Nagelsmann (35) with the team second in the standings and in the quarter-finals of the Champions League.
The win lifted Bayern back into top spot on 55 points, leaving Dortmund two points behind in second place with eight games remaining.
"There's still room for improvement," Tuchel, who coached Dortmund from 2015-17 but left on bad terms, told reporters.
"No one should be too euphoric. We are proud of the team and they can be satisfied but there's work ahead of us."
Bayern scored three times in the opening 23 minutes to kill off the game early but squandered a bagful of chances in the second half and let in two late goals.
"We gained in confidence with the first goal and led 3-0 early. I would have wished that we were more dominant after that but it was not like that," Tuchel said.
"After the 4-0 I would have wished more dominance, fewer mistakes. We had a lot of clear chances and should have led with a higher score."
Tuchel, who won the Champions League for Chelsea in 2021 after replacing Frank Lampard (44) but had been without a job since being sacked by the English side last year, admitted to a bout of nerves before the game.
"I was nervous today. I don't know why but I just was," he said. "I did not know the processes, where to find my assistants and it happened all at once and suddenly I was in the arena coaching Bayern Munich."
Tuchel, who also led Paris St Germain to back-to-back French league titles and to the 2020 Champions League final, said Bayern needed to find their rhythm with big matches awaiting and three titles up for grabs.
Bayern are back in action in the German Cup last eight against Freiburg on Tuesday before taking on Manchester City in the Champions League quarter-finals on April 11.
"We have to find a rhythm, we have to be careful, we have to improve our distances, know when to launch an attack, when to really go for it. It was a bit too hectic today."
