Thiago Cionek's own goal gave them a half-time lead in the Russian capital, with M'Baye Niang adding a second after an hour.
Niang had been on the sidelines getting treated for an injury and when the ref motioned him to return to the field, he sprinted directly towards the action with Poland's Jan Bednarik completely unaware.
Poland grabbed a late consolation as Krychowiak atoned for his earlier error by heading home in the 86th minute, however Senegal held on.
Senegal supporters celebrated their 2-1 win over Poland at the 2018 FIFA World Cup on Tuesday, June 19 by cleaning up the stadium.
Senegal had their tails up as the half drew to a close and Sane looked to double their lead from a 42nd-minute corner, but his downward header bounced up over the bar.
Poland's Wojciech Szczesny keeps the ball out.More news: European Union hits back at Trump with retaliatory trade tariffs
Senegal left the world awestruck as they produced a giant-killing in their first ever encounter against Poland, marking their World Cup start in a grand fashion.
Ranked eighth in the world, it is Poland's eighth appearance at the World Cup, with coach Adam Nawalka's side desperate to improve on the third-place finishes in 1974 and 1982.
Senegal's win means they join Japan on three points as the two became surprise leaders in Group H.
Poland will need to turn things around when it takes on Colombia - which surprisingly lost to Japan on Tuesday - while Senegal has qualification for the next phase in its sights.
"We didn't get any good forward balls to do anything with, and instead concentrated on keeping the ball at the back and keeping possession". There was Idrisa Gueye, who from midfield went on to create the shot that deflected of Cionek to open the scoring for Senegal, but it was another defensive star who cut here. Those clean sheets came against Romania and Kazakhstan - both of which are weaker opposition than what Poland will face here in Senegal.
As for Mane, he comes into the World Cup off the back of netting Liverpool's goal in the Champions League final and managed to score 20 over the whole season, half of which came in the European Cup.