Goalkeeper Alisson Becker headed a dramatic stoppage-time winner at West Brom to keep Liverpool's Champions League hopes alive.
Just as the Reds' top-four chances looked virtually over the Brazil international glanced in Trent Alexander-Arnold's corner in the fifth minute of added time to clinch a 2-1 win.
Mohamed Salah had earlier cancelled out Hal Robson-Kanu's first top-flight goal for almost four years.
Roberto Firmino hit the post and Sadio Mane had a goal disallowed for offside as relegated Albion gave as good as they got.
Kyle Bartley saw his goal ruled out as the Baggies nearly caused an upset before Alisson's late heroics.
Alisson became the sixth goalkeeper to score in the Premier League, following in the footsteps of Peter Schmeichel, Brad Friedel, Paul Robinson, Tim Howard and Asmir Begovic.
Tottenham strengthened their European qualification hopes with a 2-0 win over Wolves which moved them into the top six.
Harry Kane opened the scoring with a goal that advanced his chances of winning the Premier League Golden Boot and Pierre-Emile Hojbjerg doubled the lead after the break.
It was Kane's 22nd goal of the season and he is level with Salah in the race for a prize he has won twice before, while he also hit the post twice.
Spurs can still mathematically make the top four, but Europa League qualification is looking much more likely and a sixth-placed finish is in their own hands with two games to go.
Sheffield United teenager Daniel Jebbison scored just seven minutes into his first Premier League start to put a huge dent in Everton's hopes of European football.
The Canadian-born forward, aged 17 years and 309 days, became the competition's youngest scorer on his full debut in the 1-0 win.
His goal was enough to consign Carlo Ancelotti's side to a ninth home defeat, equalling their worst record from 1912-13, 1937-38, 1950-51 and 1993-94, with one match still to come at Goodison Park against Wolves on Wednesday.
Defeat almost certainly ended their chances of European football as they remain eighth, three points behind Tottenham and West Ham with a vastly inferior goal difference.
Tyrick Mitchell's first goal for Crystal Palace helped edge out Aston Villa in a five-goal thriller at Selhurst Park.
Roy Hodgson's side trailed twice through John McGinn and Anwar El Ghazi, but a spirited second-half rally saw the hosts recover to earn a 3-2 win.
Christian Benteke's initial leveller was followed up by goals after the break from Wilfried Zaha and another academy graduate in Mitchell, who bundled in with six minutes left to earn all three points.
