West Bromwich Albion midfielder Grzegorz Krychowiak got Poland back in the game with an 86th-minute header but Senegal held on at Moscow's Spartak Stadium.
Liverpool striker Mane set up Idrissa Gueye in the 37th minute for a shot that took a wicked deflection off Cionek, leaving Szczesny totally wrong-footed in the Polish goal for Senegal's opener on 37 minutes.
Krychowiak would get one back with a header in the 86th minute but, as Nawalka would admit after the game, it was not almost enough on a bad day at the office for the European side.
The World Cup group stage continues when Poland takes on Senegal in a Group H showdown. Poland have showed impressive performance this year since 1-0 defeat against Nigeria in March, as they registered victories over South Korea (3-2) and Lithuania (4-0), and a draw against Chile by 2-2.
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Nawalka sent on Southampton defender Jan Bednarek at halftime - Jakub Blaszczykowski making way on his 100th global appearance - as Cionek's own goal ensured that Poland's 32-year wait for a clean sheet at the World Cup goes on.
"Two goals following our mistakes was something that we were very unhappy with", Poland coach Adam Nawalka said.
By the time, Lewandowski made the first run down the middle, it was already midway through the second session (72nd) and they were already trailing 2-0. "We knew this Poland team very well", he said.
Niang had just returned to the pitch after consulting tactics with the coaching staff but the forward capitalised by getting between the two Poland players, flicking the ball towards goal and tapping into the net from close range.
Compounding the worries for African supporters is that, aside from Senegal, the continent's five teams largely aren't scoring.
"We simply lacked quality in the first half", he said.
Not only did replacement Cionek suffer the misfortune of deflecting an innocuous-looking shot into his own net, but the entire backline struggled to cope with the pace and physicality of their opponents.