Wayne Rooney came to Ronald Koeman's rescue as the Everton striker's last-gasp penalty salvaged a 1-1 draw at Brighton, while Manolo Gabbiadini's brace gave Southampton a 2-2 draw against Newcastle on Sunday.
Everton boss Koeman was on the brink of another damaging defeat after Anthony Knockaert put Brighton ahead eight minutes from full-time at the Amex Stadium.
But Brighton captain Bruno was penalised for fouling Dominic Calvert-Lewin in stoppage-time and Rooney stepped up to score the crucial equaliser.
"We are in a difficult situation but it's one thing to come out of this and show fight like we did today," Koeman said.
"At least we got one point, a point well deserved. We were mostly the better team.
"My whole football life is pressure, if I don't get pressure from outside I put pressure on myself."
While Koeman will be relieved to avoid a fifth defeat in Everton's last six Premier League games, the Toffees once again looked poor for long periods and remain without an away win since January.
The under-fire Dutchman tried to solve Everton's goalscoring problems by restoring Rooney to the attack, but until the penalty the former England captain hardly had a sniff of goal.
Koeman's frustration looked like turning to despair when Jose Izquierdo's drive rebounded into the path of Knockaert, who smashed it past Jordan Pickford to give the hosts the lead in the 82nd minute.
But Bruno's clumsy elbow into the neck of Calvert-Lewin in the penalty area earned Everton a reprieve as Rooney stepped up to roll in the spot-kick.
At St Mary's, Southampton were heading for a third straight league defeat when Isaac Hayden and Ayoze Perez put Newcastle ahead in each half.
But Gabbiadini hit back twice to celebrate his first start in more than a month.
"It was a chaotic game. We conceded a couple of soft goals. In the last 20 minutes we could win the game but we could also lose it," Southampton manager Mauricio Pellegrino said.
"As a manager I don't like this sort of game."
Newcastle boss Rafael Benitez added: "I have to be happy because we have drawn, at the same time disappointed because we could have won."
Pellegrino played under Benitez at both Valencia and Liverpool, then later joined the Spaniard's coaching staff at Anfield.
There was nothing to separate the old friends in a hard-fought clash, with the first goal coming in the 20th minute.
Christian Atsu's driven effort was blocked by Maya Yoshida, but with Fraser Forster already diving to his left, Hayden's first-time shot wrong-footed the goalkeeper.
Gabbiadini whipped the equaliser through the legs of Javier Manquillo and into the bottom corner in the 49th minute.
Perez restored the Magpies' lead two minutes later, reacting fastest to Forster saving his initial shot by squeezing in the rebound from the angle.
With 15 minutes left, Newcastle's Florian Lejeune needlessly fouled Shane Long and Gabbiadini drilled in the spot-kick.