Chelsea 1 Liverpool 0
Chelsea found a way to stop Mohamed Salah - smother him both on and off the pitch. His former Roma team-mate Antonio Rudiger was all hugs before kick-off, and then did not give the Liverpool forward an inch of space in 90 minutes as Chelsea gained the precious win that suddenly gives them an unlikely chance of qualifying for the Champions League.
Liverpool are in the final of the competition, of course, having overcome Roma in that epic semi-final last week, and would have secured their participation next season had they avoided defeat here. Hopefully they will beat Real Madrid in the final on May 26 in Kiev, but they will probably have to win at home to Brighton next Sunday to make sure they are in the Champions League next season.
Chelsea remain the outsiders for third or fourth place, with Tottenham Hotspur also in that fight. But, as Antonio Conte, the head coach, said, "we are still alive" and this could go down to the wire. It might even be decided on goal difference and that is also against Chelsea.
A Champions League place would make Chelsea a far more attractive proposition for whoever succeeds Conte, with the Italian still expected to leave Stamford Bridge in the summer, even if he also wins the FA Cup.
Liverpool's physical and emotional exertions in overcoming Roma, and the fierce heat on this occasion, were undoubtedly factors, as was the shallowness of the bench that Jurgen Klopp, the manager, was able to call upon. Through injuries, the midfield is looking threadbare and this proved a difficult afternoon for his full-backs, Nathaniel Clyne and Andrew Robertson, who were both substituted.
Even so, Liverpool claimed 68 per cent of possession and, at times, were camped deep in the Chelsea half, although that also felt like a deliberate tactic from Conte, who set up his team to drop deep, deny their opponents space, and defend with aggressive discipline - which was the way Rudiger handled, quite literally at times, Salah.
It was the first time, in fact, that Liverpool had failed to score in back-to-back league games under Klopp and their cause was not helped by a yellow card for Salah for diving, as he threw himself to the turf expecting a challenge from Gary Cahill. It was Salah's first booking of the season and it seemed to blunt him, fearful he might be booked again if he reacted to Rudiger's attentions.
The frustration for Liverpool was increased by the nature of Chelsea's goal, which Klopp later lamented came from nothing more than a routine cross, although that was a little harsh on Tiemoue Bakayoko, who had brilliantly switched play with a precise cross-field pass to Victor Moses, out wide on the right. The wing-back cut inside, with his cross clipping off Sadio Mane. Even so Liverpool - and, again, it felt the finger was pointed at Dejan Lovren - should have dealt with it, but instead Olivier Giroud rose to cleverly guide his header across goalkeeper Loris Karius and high into the net. Trademark Giroud.
The celebrations involved the striker sprinting across to the bench and hugging injured team-mate David Luiz and, out of the blue, Chelsea had something to cling on to which, given their tactics, made Liverpool's task even more difficult. Before the goal, Liverpool had chances and maybe should have scored but they just could not get Salah involved. Instead, it was Eden Hazard who gradually became the game's dominant attacker. At one stage his shirt was ripped across his chest, before he donned a fresh one, but Hazard shrugged that off as he danced and wriggled his way through challenge after challenge.
There was one extraordinary movement when he tricked a route past three Liverpool defenders through the tightest space and Karius turned his cross-shot away.
Liverpool desperately wanted the draw, to end this race, and re-doubled their efforts. Klopp re-shaped his team and introduced a fourth forward in Dominic Solanke. Like Salah, the 20-year-old is a former Chelsea player and maybe he would provide the twist, and not his feted team-mate, as he attempted to score his first Liverpool goal. Solanke came close. Twice.
He had two headers, with the second sailing over in injury-time to consign Liverpool to defeat and give Chelsea hope of at least taking the top-four race into the final weekend - providing they beat Huddersfield Town in midweek - when they are away to Newcastle United. Spurs host Newcastle on Wednesday, before welcoming Leicester City.
Liverpool have bigger targets and, as Klopp noted, at least have some time to rest and recuperate before the first of their two "cup finals" (Brighton then Real Madrid). The frustration is that, had they not lost here, it would have just been one final. The frustration for Chelsea is they have not produced enough smart performances like this throughout the season.