Back-to-back home losses had left the Cubs on the brink of defeat in the best-of-seven series, but Sunday's win means they trail 3-2 as they return to Cleveland for Tuesday's Game Six.
Kris Bryant's fourth-inning home run proved decisive as the Cubs claimed a first home World Series win since 1945.
The Cubs have gone 108 years without winning baseball's biggest prize.
The Indians last won in 1948 and can end that drought - the second longest in baseball behind the Cubs - with victory on Tuesday.
Game Seven, if required, will also be played in Cleveland, on Wednesday.
The Chicago Cubs kept their World Series hopes alive with a 3-2 win over the Cleveland Indians at Wrigley Field.