Mercedes driver Hamilton began proceedings at Hockenheim in 20th place due to a crash in qualifying that left him 15th and a five-place grid penalty for changing his gearbox.
However, despite that disadvantage, the Briton put on a display of ruthless attacking driving to charge up the field.
Within the first quarter of the grand prix, Hamilton had moved into the podium places, before engaging in an enthralling battle with Williams' Valtteri Bottas for second behind his team-mate and championship leader Nico Rosberg.
The 2008 world champion made up sufficient ground to attack Bottas in the closing stages, but the Finn stood firm to deny him.
Rosberg extended his lead at the top of the drivers' standings to 14 points with a comfortable start-to-finish victory, but Hamilton was in upbeat mood after the race.
"I had great fun," said the 29-year-old, who collided with McLaren's Jenson Button on the 30th lap and suffered a damaged front wing.
"I did as good as I could. It was hard to get through the pack safely and I had a little bit of a collision with Jenson.
"I thought he was going to open the door, which he has done a couple of times lately, but that was just my bad judgement.
"It was hard to overtake so I'm glad to get some points today."
Bottas, meanwhile, was thrilled to have resisted Hamilton's challenge on his way to claiming a third successive podium finish.
"It was not easy and required input from all the engineers," he said.
"Thanks to all the fans. I saw many Finnish flags so thank you."
Rosberg is back in full control of the race for the title and he could not hide his delight at winning his home grand prix.
"It's fantastic," said the German.
"It's an amazing feeling to win at home. It's a very special day."