Following an early flurry from the home team at Signal Iduna Park, during which time Marco Reus rattled the crossbar with a 16th-minute free-kick, the Bundesliga's bottom team settled into a defensive rhythm to frustrate Dortmund.
On the back of a goalless draw at Hertha Berlin last weekend, Dortmund were anxiously contemplating losing further ground to champions Bayern Munich at the summit before Henrikh Mkhitaryan's superb solo effort settled matters in the 57th minute.
"It was tough today," the Dortmund boss told a post-match press conference. "We started very well in the first 15 to 20 minutes.
"We won the ball a lot and had great scoring chances, Marco was almost perfect with that free-kick that hit the bar.
"After that, we were not so precise and more distracted because of technical errors."
Tuchel was without Pierre Emerick-Aubameyang after his top scorer suffered a knock during the midweek DFB-Pokal win over Stuttgart and gave credit to Hannover for pushing them close.
"Hannover also defended well," the trainer continued. "We on the other hand found no room and, when we did, we did not find the final ball.
"We can do much better and have room for improvement.
"Miki [Mkhitaryan] then ended a counter-attack spectacularly. He had to decide late but hit the ball cleanly. We are pleased about the result."