On the back of 1-0 victories over Latvia and Slovakia during their training camp in Austria, the tournament co-hosts gave themselves an easy ride on home turf. French-born Bordeaux midfielder Obraniak put Poland ahead in the 13th minute after latching onto a lateral pass from Terek Grozny's Maciej Rybus.
With the bulk of the action in Andorran territory, a foul on Lewandowski by midfielder Marcio Vieira - one of the few to ply his trade abroad, in the Spanish third division - led to a heated exchange six minutes later.
Rybus nearly made it two, but fired his free-kick too high, before Lewandowski received the ball from Obraniak, only to head over the bar.
Poland manager Franciszek Smuda, at the helm since October 2009, traditionally relies on players from foreign leagues, notably Germany's Bundesliga. They include striker Lewandowski, midfielder and captain Kuba Blaszczykowski and defender Lukasz Piszczek, all at German double winners Borussia Dortmund.
Continuing to harry Andorra goalkeeper Josep Gomes, who also plays in Spain's third tier, Poland doubled their tally in the 37th minute as Lewandowski met an aerial pass from Piszczek with a bicycle kick that found the net.
Two minutes later, Poland were awarded a penalty after Blaszczykowski was brought down in the box and he fired in from the spot to make it 3-0.
Andorra's best chance came in the second half, after a 62nd-minute foul by substitute defender Grzegorz Wojtkowiak of Lech Poznan, one of the few Polish players to play domestically. But Arsenal keeper Wojciech Szczesny was untroubled as the subsequent Andorran free-kick went wide, and Legia Warsaw youngster Rafal Wolski immediately forced Gomes to save a counter-strike.
Having failed to reach the 2010 FIFA World Cup™, and with an automatic hosts' berth at EURO 2012, Poland will have gone 968 days without a competitive match by the time they kick off against Greece in the tournament's opening game on 8 June in Warsaw.