Sterling opened the scoring after Kevin de Bruyne's beautifully weighted pass saw his team-mate outsprint Patrick van Aanholt before an exquisite finish beyond Vicente Guaita.
England winger Sterling's instinctive finish after Leroy Sane's pass doubled the lead before Palace, who are 11 points above the relegation zone, reduced the deficit through Luka Milivojevic's late free-kick.
Guardiola stated afterwards, "This pressure is incredible. We are fighting to be champions back to back. In two seasons we've got 83 points with five games left. We've won two titles (Community Shield) and now it's the FA Cup final.
On Sterling's two goals, he added: "He missed the first chance but the first goal was incredible. He's a guy who has scored a lot of goals. He fought and helped Kyle Walker."