Having turned 23 last Wednesday, Podolski combined with Germany's other Polish-born forward Miroslav Klose for his first goal on 20 minutes and followed up with an unstoppable volley on 72 minutes.
After Group B rivals Croatia had earlier got off to a winning start with a 1-0 victory over co-hosts Austria in Vienna, the result puts the Germans top of their group.
Podolski admitted he had somewhat mixed feelings after sending his native land packing. "We did the essential in winning our opener, but this victory does not over enthuse me as I am originally from Poland. All my family is over there. I have two hearts - a German one and a Polish one!"
The last time Germany had won a European Championship game was way back in June 30 1996 when they beat the Czech Republic to win Euro 1996 at Wembley.
There had been some controversy last Wednesday when Polish tabloid Super Express printed a mocked up picture of Poland coach Leo Beenhakker holding the severed heads of opposite number Joachim Löw and Germany captain Michael Ballack. Both sides had played down any problem, but 140 Germans were arrested here in the city centre on Sunday.
A first group of about 60 was arrested after they chanted racist slogans, including "all Poles should have to wear yellow stars," referring to the insignia the Nazis forced Jews to wear before and during World War II. A few of the hooligans managed to run away but the majority were taken into custody.
Shortly after the start of the game, another group of about 40 men this time appeared, chanting similar Nazi slogans such as "German fight back. Don't buy from Poles," the Austria Press Agency reported. They too were arrested and taken away in police vans.
But the proper football fans in the stadium were well-behaved and they lent their passionate support during the match.
Ballack said the Germans had been their traditional efficient selves in sealing the points. "I think we played well. It's always difficult - you never know where you are before the first game of a tournament. I think we deserved victory. From the start we were aggressive," the Chelsea star told the BBC.
Polish losing streak
The Poles were making their European Championship debut, but are still yet to beat Germany in 16 meetings between the sides since 1933 with the Germans now claiming 12 wins with four draws between the neighbours.
Beenhakker had opted for five in midfield with Euzebiusz Smolarek as the lone striker up front for Poland. Löw had gone for a traditional 4-4-2 formation, keeping midfielder Bastian Schweinsteiger - a star of the 2006 World Cup - on the bench until half-time playing Podolski behind the two German strikers.
The move paid off as Podolski gave Germany an extra attacking option supporting front men Miroslav Klose, the top scorer at the last World Cup, who was partnering VfB Stuttgart hot-shot Mario Gomez up front.
After both sides squandered early chances, the trio combined to give Germany the break through they craved on 20 minutes. Gomez flicked the ball over the defence for Klose to draw the goalkeeper before giving his Bayern Munich team-mate Podolski a simple tap in on 20
minutes past Poland's Celtic goalkeeper Artur Boruc.
Poland's best chance of the first half - and ultimately the game - came when midfielder Wojciech Lobodzinski put in a superb pass, but Maciej Zurawski pulled his shot wide. Polish striker Smolarek picked up the first booking on 40 minutes when he tripped Per Mertsacker.
During the half-time break, Beenhakker swapped his captain Maciej Zurawski for Brazilian-born Roger Guerreiro, but he made little impact and Lobodzinski earned Poland's second yellow card on 64 minutes for tripping Podolski. Germany were dominating by this stage and when the ball fell to Podolski on 72 minutes he drove his shot into the top corner of the net to give his side some breathing space.
German goalkeeper Jens Lehmann produced a string of saves as Poland applied some late pressure, but his defence held. The result means Group B's two unbeaten sides will meet Thursday when Germany return here to face Croatia.
Ballack said he was looking forward to playing Croatia but cautioned: "I think it's an open game - both teams are very strong."