The Cubs put runners on second and third with two outs in the ninth before Allen struck out Javier Baez to end it.
Pinch-hitter Coco Crisp hit an RBI single in the seventh off Carl Edwards Jr. for the lone run.
Indians starter Josh Tomlin went 4 2/3 innings, then Andrew Miller, Bryan Shaw and Allen took over.
The Cubs have been blanked four times in the last eight games this postseason. It was their first 1-0 loss in the World Series since Babe Ruth and the Boston Red Sox beat them in 1918.
Game 1 winner Corey Kluber starts for Cleveland on Saturday night on three days' rest, and John Lackey goes for Chicago.
For the first time since 1945, the Chicago Cubs played in a World Series game at Wrigley Field Friday night.
The Cubs announced the team's starting lineup for Game 3 on Twitter Friday afternoon.
Billy Williams, Baseball Hall of Famer, Cubs legend and six-time MLB All Star, threw out the ceremonial first pitch at Game 3. Bill Murray, actor and lifelong Cubs fan, sang the 7th Inning Stretch. Fall Out Boy lead singer and Chicago native Patrick Stump sang the National Anthem. Wayne Messmer, who usually sings the National Anthem, performed "God Bless America" during the pre-game ceremonies.
"I'm going to the game tonight with my son, Hank. I'm just giddy. Just giddy about it," said Joel Murray, Bill Murray's brother.
WORLD SERIES TICKET PRICES
Dozens of people bundled in jackets and hooded sweatshirts lined up outside the ballpark box office, hoping to get last-minute tickets to the historic game.
The 2016 World Series yielded historically high ticket prices on the secondary market.
StubHub's Cameron Papp from joined ABC7 News at 11 a.m. to explain how this year's World Series compares to last year, how demand for these tickets compares to other major events and how much fans will have to shell out to watch the next three games.
Game 3 tickets range from $1,200 and $19,000 and tickets for Games 4 and 5 range from $1,100 to $18,000, Papp said.
BEHIND-THE-SCENES AT WRIGLEY FIELD
ABC7 Meteorologist Tracy Butler gave fans a view of the Friendly Confines that not many people get to see.
WRIGLEY FIELD GRUB
The food offered inside the ballpark has changed dramatically over the years. It is no longer just a beer, a hot dog and a brat.
Chef Ron Krivosik from Levy Restaurants joined ABC7 News at 11 a.m. to share a sampling of Wrigley Field's World Series eats.
The last time the Cubs won a World Series was in 1908. Tickets to the World Series only cost about $1.20, which is equal to about $40 today. This year, tickets are about 1,000 times as expensive.
The last time the Cubs were in the World Series was 1945. That was 10 years before Richard J. Daley became Chicago mayor, and two years before Jackie Robinson made his Major League Baseball debut. Dexter Fowler was the first African-American Cubs player to start in a World Series.
Wrigley Field is also a lot different than it was the last time a World Series game was hosted at the ballpark. Much has changed in the last year alone. Massive renovations include a new clubhouse, new bullpens and a new upper-level concourse.
WORLD SERIES 2016 SCHEDULE
Game 3 will start at 7 p.m. Friday in Chicago.
Game 4 will start at 7 p.m. Saturday in Chicago.
Game 5 will start at 7 p.m. Sunday in Chicago.
If necessary, Game 6 will start at 7 p.m. Tuesday in Cleveland.
If necessary, Game 7 will start at 7 p.m. Wednesday in Cleveland.
CLICK HERE for more information on security guidelines, public transportation, street closures, parking restrictions in Wrigleyville during the World Series.
The Associated Press contributed to this report.