The 23-year old striker joined the Hoops from English League One side Wolves for around £1million on deadline day and netted his first goal for the Glasgow giants in Saturday's 2-0 victory at Heart of Midlothian.
Lennon witnessed Griffiths score over 30 goals in two loan spells with Hibernian over the past few seasons and he is now expecting him to repeat that sort of form for the Bhoys as they chase a third straight Scottish title.
"He's settled in fine," said Lennon. "Getting the goal will have given him a huge psychological lift.
"He has been brought here to score goals and there is an expectation on him now. He's playing for the biggest club he's had in his career now but he's adapted to it very well and the players around him have made the transition easier for him.
"It will be a weight off his shoulders to get that goal so early in his time here. He's a goalscorer and if you look at his finish, it was a typical Leigh Griffiths finish.
"He strikes the ball with so little back-lift but with a lot of power and unerring accuracy. We like his movement and his intelligence and I think there is more to come from him."
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