Duvan Zapata scored for the second match running to send Colombia into the Copa America quarter-finals with a late header to beat Qatar 1-0 on Wednesday.

Carlos Queiroz's team became the first side into the knock-out rounds after following up Saturday's 2-0 win over Argentina with another victory over the dogged Asian champions.

It was a deserved success but one that had looked likely to elude the 2001 champions as they toiled against a Qatar side that defended stoically and held on grimly for more than 85 minutes.

"It was difficult, but we showed patience and mustered the winning goal. The win was paramount and we leave happy," said Zapata, who insisted Colombia wouldn't be taking things easy in their final group match against Paraguay on Sunday.

"We hope for the best possible finish to the group stage."

Qatar, who will host the next World Cup in 2022, may now need to beat Lionel Messi's Argentina in their final Group B match if they are to reach the last eight.

"It leaves a bad taste because the end was close," said Qatar's Spanish coach Felix Sanchez.

"Conceding a goal in the final minutes when the game was heading for a draw ... it leaves you bitter at not having taken a point but we also have to appreciate that the team competed with great opponents."

Zapata repaid Queiroz after the Portuguese coach surprisingly picked the Atalanta striker in place of star forward Radamel Falcao.

"We are on the right path. When you have Falcao and Duvan in your side, scoring is a given," said captain James Rodriguez.

Queiroz's team was off to a flying start, putting the ball in the net after just six minutes, but Roger Martinez's close-range header was ruled out for offside.

James should have given Colombia the lead on 13 minutes as he met Martinez's pinpoint cross just seven yards out, but the playmaker inexplicably headed wide.

Much as they had been in the early stages against Paraguay, when they came back from 2-0 down to draw 2-2, Qatar were under the cosh as Colombia spread the play intelligently.

Juan Cuadrado tried his luck from 25 yards but goalkeeper Saad Al-Sheeb was alert to punch the ball away.

Zapata, as he had during the last Serie A season, caused havoc in the Qatar defence with his size and strength but was off target with a pair of headers.

At times it looked like a game of attack versus defense but Colombia were struggling to break down the massed ranks of burgundy jerseys.

When Martinez did split open the defence with a through ball for Cuadrado, the Juventus winger's first touch was too heavy and Boualem Khouki darted across to clear the danger.

Colombia thought they had a penalty early in the second half but VAR -- after an interminable delay -- overturned referee Alexis Herrera's handball award against Abdulaziz Hatem.

James found Martinez inside the area with a clever pass but the winger fired straight at Al-Sheeb.

Qatar started to show greater ambition and Colombia's goalkeeper David Ospina was called upon to make low saves from Hatim and Akram Afif.

Queiroz reacted by bringing on Falcao for Cuadrado and moments later James teed up Martinez again, but the Club America forward was denied by Al-Sheeb.

Colombia kept creating chances but at times they simply pumped aimless crosses into the Qatar box.

The breakthrough finally came four minutes from time as James dinked a chip with the outside of his left foot to the back post where Zapata rose highest to power home a header.

"They said we couldn't play together with Falcao, but we showed that we could play with two strikers," said Zapata, who lept above his strike partner to convert the winner.

He should have had a second in injury time but fired wide on the counter-attack with only Al-Sheeb to beat.