Zapata rose with five minutes left to head home a floated cross from Man of the Match James Rodriguez and break the deadlock in a scrappy but end to end encounter at the Morumbi stadium.
The result means Colombia, who beat Argentina 2-0 in their opening match, now have six points and top Group B, with Argentina playing Paraguay later on Wednesday.
Qatar have one point after drawing 2-2 with Paraguay on Sunday but could still qualify for the last eight.
Colombia were the better team in the first half, with Roger Martinez having a goal ruled out for offside after just five minutes and Rodriguez putting a free header wide eight minutes later.
Asian champions Qatar, one of the two non-South American teams invited, improved in the second period but just as the game looked destined to finish goalless, Zapata gave Colombia all three points.
"We’re on the right track but we’re still a long way from achieving all we really want," Rodriguez told reporters.
A poor crowd of just over 22,000 people watched the game on a chilly night.
The final games in the group take place on Sunday when Colombia face Paraguay and Qatar play Argentina.
(Editing by Ian Ransom)
