The teams will meet at 11 a.m. ET on Tuesday in their 2018 FIFA World Cup Group H opener.
Piszczek then failed to hit the target on the volley after connecting with Maciej Rybus' cross from the left on 56 minutes, before a mix-up at the back gifted the Senegalese with a second goal soon afterwards.
The win keeps Senegal's record in opening World Cup matches at 2-0, their first famously coming over holders France in 2002.
The Asian team overcame Colombia by the same score earlier in the day.
Senegal: Khadim N'Diaye, Wague, Sane, Koulibaly, Sabaly, Sarr, Alfred N'Diaye, Gueye, Niang, Mane, Diouf.
Poland advanced from the group stage in four consecutive World Cups between 1974 and 1986, and they earned at least a draw each of the times. They were unable to get the equalizer, however, and Senegal took the match 2-1.
"I don't know if you get an award for that, Jon, but it's something Federation Internationale de Football Association should look into", he said with full sincerity.
But it hardly matters who is officially awarded what; Mane proved his importance by playing a key role in Senegal's attack without really standing out.
Lewandowski topped the Bundesliga in scoring in three of the last five seasons and while he had 16 of Poland's 28 goals in qualifying, he managed only one goal at the 2012 European Championship and one at Euro 2016.
Instead, the speedy Mbaye Niang got in front of his defender, nicked the ball away from Szczesny and rolled it into the empty net. That's when you need your best players to produce small moments to edge the contest, and Mane subtly stepped up.
Senegal is making its second World Cup apperance ever.
Dawid Kownacki also sent a header too close to N'Diaye as the match edged closer to full time but with four minutes left, Poland finally got themselves back into the contest through Krychowiak, who powered a header into the far corner after meeting a deep set piece.
But while their two goals carried more than an element of fortune, Senegal were much the better side, looking controlled in midfield and more dynamic in attack, and that's what appeared to irk the Bayern Munich forward the most.