Barca took the lead in the 13th minute as Kessie nodded the ball home after Jules Kounde had headed Ousmane Dembele's corner back across goal.
It was the Ivorian midfielder's first goal for the club since signing from AC Milan in the close season.
Ansu Fati almost doubled the advantage midway through the first half, but his effort was saved by visiting goalkeeper Jindrich Stanek.
Plzen were awarded a penalty in the 25th minute as Andreas Christensen brought down Jhon Mosquera, but the referee changed his mind after a VAR review and instead gave a free-kick to Barca.
Lewandowski moved ahead of Karim Benzema into third place on the all-time Champions League goal-scoring charts with his 87th in the competition, and first for his new club.
The 34-year-old curled a low shot into the bottom corner from just inside the area in the 34th minute.
The Czechs gave themselves a lifeline shortly before the break as Jan Sykora headed in Vaclav Jemelka's cross.
But Plzen could not stay within a goal until half-time as Lewandowski stooped unmarked to head Dembele's cross over the line.
Lewandowski completed his sixth Champions League hat-trick with a fine low strike from distance in the 67th minute.
Xavi's men wasted little time in making it five, as substitute Ferran Torres drilled a brilliant volley on the turn into the top corner from Dembele's chipped pass.
Plzen avoided any further damage but face a daunting task in their remaining five group matches.