Kluber struck out eight of the first nine outs registered and Roberto Perez hit two home runs as the Cubs endured a miserable return to the season finale.
The Cubs' last World Series win was in 1908, while the Indians clinched their most recent title in 1948.
Game Two of the best-of-seven series begins at 00:08 BST on Wednesday.
The first pitch has been brought forward an hour because of the threat of rain in the Cleveland area.
A superb pitching display from Corey Kluber helped the Cleveland Indians to an emphatic 6-0 win over the Chicago Cubs in the World Series opening game.