The finale may be how Liverpool like it – an all-or-nothing, death-or-glory Champions League decider at Anfield – but not how they want it. Jürgen Klopp lamented Neymar’s antics, the referee and Paris Saint-Germain gamesmanship but ultimately had only Liverpool’s lacklustre start to blame for another European away defeat. “A big step forward for us,” beamed Thomas Tuchel. Liverpool are in danger of finding reverse in the competition they cherish.

Inspired by the infuriating but sometimes unplayable Neymar, PSG delivered a deeply impressive performance that finally lived up to their expensive ambitions among the European elite. They lost their shape and composure once James Milner halved a deficit delivered by goals from Juan Bernat and the world’s most expensive footballer but there was no disputing the merit of a victory that the French champions had to deliver.

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Liverpool pressed for an equaliser in the second half but in vain, failing to register a shot on target other than Milner’s penalty in first-half injury time. Defeated in all three away games for the first time in the Champions League group stage, Klopp’s team must beat Napoli 1-0 or by a two-goal margin on 11 December to avoid early elimination. It is not beyond Liverpool by any means, and Carlo Ancelotti knows only too well the powers of their recovery, but a vastly improved European display will be required to avoid the rigours of the Europa League in the new year. The visitors’ midfield was bypassed regularly here by the speed and skills of Neymar and Kylian Mbappé. Their threat to Gianluigi Buffon in the PSG goal was virtually non-existent. Poor decision-making was not the sole preserve of the Polish referee Szymon Marciniak, whose qualifications for a fixture of heavyweight calibre were called into question by Klopp.

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Tuchel had conceded that PSG have a mental difficulty in switching from dominating in France, where they have won the last 14 domestic cup competitions and five of the last six league titles, to imposing themselves in the Champions League. Not here. His team started superbly and the problems in Paris belonged exclusively to those in red.

The PSG coach, said Klopp, would have been working on a plan to avenge the 3-2 defeat at Anfield within seconds of Roberto Firmino’s winner. It materialised in a surprising 4-4-2 formation with Ángel Di María on the right and Neymar left. Not only did it afford the Brazilian space to wreak havoc against Liverpool’s pedestrian three-man midfield, it enabled Mbappé to preoccupy the visiting defence with his pace. With Marco Verratti commanding in central midfield and PSG a whirl of white as they opened at blistering speed, Liverpool were besieged from the off. There was absolutely no surprise inside the raucous Parc des Princes when they fell behind with only 13 minutes gone.

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Neymar breaking from deep and Mbappé skipping clear of Joe Gomez and Dejan Lovren was a recurring theme of the first half. The breakthrough stemmed from Mbappé’s low cross from the left. Virgil van Dijk scuffed his attempted clearance skywards, Gomez and Lovren were slow to react and Bernat steered a low finish beyond the wrong-footed Alisson.

Verratti was dictating proceedings with style and far too much freedom but was fortunate to escape a red card for a late and high tackle that left Gomez writhing in agony. A yellow card was deemed sufficient, and the midfielder also sailed close to a second when clipping Jordan Henderson. Liverpool improved slightly after the opening goal but were still frequently overrun in midfield. They fell two behind when the Neymar-Mbappé combination exposed their midfield again. The Brazilian sent Mbappé scampering down the left and, though Alisson made a fine block to deny Edinson Cavani converting another low Mbappé centre, the rebound rolled perfectly for Neymar who made no mistake from six yards. The kick-off was delayed by his over-elaborate celebrations.

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Liverpool were passive but given hope when Di María conceded a penalty with a rash challenge on Sadio Mané. Mané invited the tackle superbly, beating Marquinhos deep in the PSG half before embarking on a piercing run that evaded one defender before Di María sent him sprawling just inside the area. The referee initially, bizarrely, awarded a corner only to be corrected by the additional assistant behind the goal. Milner sent Buffon the wrong way from the penalty spot and Liverpool, from out of nowhere, were back in the contest.

PSG bemoaned the penalty decision all the way down the tunnel at the interval. One coach pursued the additional assistant referee with an iPad showing a replay of the incident. Their chaotic defending after the restart illustrated the destabilising effect of the Liverpool goal and offered Klopp’s team several chances to restore parity. But despite Liverpool regularly outnumbering PSG defenders Buffon was never tested. Firmino flashed a header wide and Mohamed Salah had a shot blocked on the edge of the area. The home crowd were growing restless, their anxiety not helped when Marquinhos had a goal disallowed for offside when converting a free-kick moments after the restart.

Facebook Twitter Pinterest Liverpool’s James Milner scores from the penalty spot past Gianluigi Buffon of PSG. Photograph: Andrew Boyers/Action Images via Reuters

The converted central midfielder was also denied by Alisson when he met Neymar’s corner with a towering header that the Brazilian keeper saved well. Andy Robertson stopped Mbappé with a superb challenge as he shaped to shoot inside the area as PSG finished confidently. Liverpool have it all to do on the final day, yet again.