LOS ANGELES : Stephen Eustaquio's stoppage-time winner fired Canada to a 1-0 win over South Africa on Sunday in the first match of the World Cup knockout rounds, as the co-hosts advance to the last 16 for the first time in their history.

With both sides making their inaugural appearance in the knockout phase, the game appeared to be heading for extra time, before Eustaquio broke South African hearts with a clinical hit from just outside the area.

Canada -- playing in Los Angeles despite being co-hosts, after failing to win their group -- will now face the Netherlands or Morocco in Houston in the last 16 on July 4, as South Africa's World Cup adventure comes to an end.

"We felt like we could wear them down over time, and it took 92 minutes, but we got there," said Canada coach Jesse Marsch.

The manager said his team could have "made life a little easier on ourselves" by converting multiple good chances earlier in the game.

"But obviously the timing of the goal means that the win is incredibly dramatic, and I think the effect that it will have in Canada and the inspiring of people will be immense," he added.

Before the game, Marsch had talked up the "X-factor" of Canada's star Alphonso Davies, available again after a hamstring injury.

But the Bayern Munich wing-back, who was yet to appear at this World Cup, remained glaringly absent from the starting line-up.

Still, Canada began on the front foot, racking up chances against South Africa's low block.

In the 22nd minute, a pinpoint Eustaquio free-kick found Derek Cornelius wide open on the six-yard line. The Canada center-back utterly mistimed his headed effort.

South Africa threatened sporadically on the counter. But in a frenetic period just before half-time Canada came agonizingly close.

From a corner, Moise Bombito's header was cleared off the line, before Tajon Buchanan's follow-up shot was blocked by goalkeeper Ronwen Williams.

Moments later Khuliso Mudau crashed into the dangerous Richie Laryea, but his penalty appeals were waved away.

That enraged a heavily Canadian crowd, and the jeers grew louder after half-time as South Africa were content to exchange deep passes under little pressure while the clock ticked down.

Just past the hour mark, Canada came close again.

Substitute Niko Sigur slid a through ball to Tani Oluwaseyi whose shot was well-saved by the goalkeeper and looped up toward Jonathan David -- but defender Mbekezeli Mbokazi just got there first and cleared brilliantly.

- Davies returns -

In the 75th minute Davies finally appeared, to a huge roar, and made an immediate impact.

From the left wing, Davies found Jonathan David who laid the ball to fellow striker -- and no relation -- Promise David, but his shot went wide.

Moments later Davies found Jonathan David again. This time he shot from a tight angle, but was denied by Williams.

In the second minute of stoppage time, Canada finally broke through, from the other flank.

Jacob Shaffelburg raced down the right wing and pinged in a cross which was headed clear to Eustaquio.

The Porto midfielder -- on loan at Los Angeles FC -- chested the ball down and slammed it first-time into the bottom-left corner.

Canada's bench streamed onto the pitch as the stadium erupted, celebrating a historic victory.

"It's a shame we couldn't do it in Vancouver, in front of our fans," said Marsch.

"But nonetheless, I think you saw the character of the team, the quality of the team, the mentality of the team, the togetherness of the team."

South Africa coach Hugo Broos said his defeated side "can be fairly satisfied with what we did" by progressing from the group stage for the first time.

"Okay, we are disappointed, we wanted to win, but it would have really been a little miracle if we should have been in the third round," he said.