Dortmund entered the match on the back of successive home defeats but warmed up for Tuesday's Revierderby against Schalke with an excellent triumph on the road.
Mats Hummels scored just before half-time for Dortmund, who were not at their best before the break but improved significantly after that, with Robert Lewandowski's outstanding solo goal the highlight.
Marco Reus also netted a late strike as Klopp's second-placed side made sure that Bayern Munich had to wait to claim the Bundesliga title.
The nature of the victory – hard-fought and gritty – delighted Klopp, who was happy to see his side respond to a slow start.
"It was a very tough game," Klopp said.
"Two teams had to work hard. We had to do without some key players and Hannover also had to do without many important players.
"We were able to play in the first half, control without (dominating), and make a standard situation (into) the first goal.
"I was very happy because we are working hard on such things.
"(At) 2-0 we turned in the right direction. From then on it was very difficult for the opponent."
Meanwhile, Klopp hailed as "outstanding" and "a world-class goal" Lewandowski's strike, which saw him collect the ball on the halfway line and cut through Hannover's defence with ease.
Hannover coach Tayfun Korkut paid credit to Dortmund and felt his side had failed to deal with the visitors' gameplan.
"It was a deserved victory for Dortmund. It was not good enough," Korkut said.
"We knew that Dortmund (would) not (be) quite as aggressive… but would rather wait for a chance. This has hurt us."
