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Liverpool’s Alexis Mac Allister headed home a second-half winner as the team earned a hard-fought 1-0 victory over Real Madrid in a battle between European giants at Anfield, with Madrid grateful to goalkeeper Thibaut Courtois for keeping the score down, Aljazeera reported.

Trent Alexander-Arnold was met with a hostile reception on his return to Liverpool as a late substitute on Tuesday, and the home crowd roared the Reds over the line for what could be a turning point in their season.

Both sides are now tied on nine points from their opening four games of the league phase, according to Aljazeera.

Luis Diaz scored twice and was sent off as Bayern Munich maintained their perfect record this season with a statement 2-1 win away to title-holders Paris Saint-Germain on Tuesday in a glamour Champions League tie marred by a first-half injury to Achraf Hakimi, Reuters reported.

The Colombian gave Bayern a fourth-minute lead at the Parc des Princes and then pounced on slack defending to score again just after the half-hour mark.

However, the former Liverpool winger was then shown a red card in first-half stoppage time following a rash challenge on Hakimi which forced PSG's Moroccan star off in tears, according to Reuters.

Atletico Madrid got a 3-1 home victory against Union Saint-Gilloise in their Champions League clash on Tuesday, with goals from Julian Alvarez, Conor Gallagher and Marcos Llorente keeping Diego Simeone's side in the hunt for qualification, Reuters reported.

The result leaves Atletico 14th in the league phase standings on six points from four matches, level with five other teams, while Belgium champions Union Saint-Gilloise sit 26th with three points and outside the qualification spots.

Union Saint-Gilloise started aggressively, exploiting counter-attacks with speed, but Atletico struck first in the 40th minute when Giuliano Simeone burst up the right channel and put it on a plate for his Argentina team mate Alvarez to unleash an unstoppable half-volley from inside the box, according to Reuters.

Juventus are still without a win in the Champions League after they were held to a 1-1 draw in Turin by Sporting on Tuesday, when Maximiliano Araujo put the visitors ahead and Dusan Vlahovic equalized for the hosts, Reuters reported.

An eight-game winless streak in all competitions led to the recent sacking of Juventus coach Igor Tudor and, after two victories on the bounce since then, another draw in Europe, this time under Luciano Spalletti, put the Italians on three points from four matches.

"We are not happy with the result, we could have done more," Vlahovic told Sky, according to Reuters.