Everton's first derby victory at Goodison Park in more than 13 years prevented Liverpool from climbing level with Arsenal at the top of the league table. Juergen Klopp's men are three points behind the Gunners (77) with four games to play.
Holders Manchester City remain in the driver's seat on 73 points but with two games in hand, while Everton's victory has them breathing easier, eight points above the relegation zone in 16th.
"Very disappointed in so many ways and I think everyone has to look in the mirror look at their own performances, did they give everything and do they really want to win the league?" Liverpool captain Virgil van Dijk said.
"Obviously there are still games after tonight, but if we play like we did tonight, we have no chance to win the title. It's a tough one and obviously we have to do much better against a side that is playing against relegation."
Everton were sharper from the start and won a penalty when Calvert-Lewin was tripped by Liverpool keeper Alisson only for it to be overturned by the VAR for an offside.
Yet it wasn't long before Branthwaite had the old stadium shaking in the 27th minute when he capitalised on poor defending from Liverpool, slotting home from close range after the Reds missed several attempts to clear the ball.
Alisson could not quite get his hands on the ball which rolled off the post and in.
Branthwaite is the first Everton player to score against Liverpool since Demarai Gray in a 4-1 league home loss in late- 2021.
"We spoke about the desire and the energy of the team and the physical side of the game," Everton manager Sean Dyche said. "The lads did it on Sunday (2-0 win over Nottingham Forest) and they'll have to do it again. That's the challenge.
"Sunday was massive and now it was a massively important derby game."
Calvert-Lewin leapt to head home from a corner kick to double the Toffees' lead in the 58th minute, and while Liverpool have mounted some masterful comebacks this season, Everton held strong to hand Klopp, who is leaving his role at the season's end, his first defeat at Goodison in his final derby.
"It's extra special, isn't it? Local derby -- I've been here long enough to know what it means to the fans, what it means to me," Calvert-Lewin told Sky Sports. "Where we are in the league, we needed a massive three points and everything kind of came together tonight."
Liverpool kept Everton keeper Jordan Pickford busy in the dying minutes as he leapt to bat Harvey Elliott's shot over the crossbar and then dived to save a shot from Reds talisman Mohamed Salah while Everton manager Sean Dyche looked at his watch in frustration, desperately wanting the final whistle to blow.
It was joyful bedlam when it finally did, with Everton's faithful serenading the team off the pitch.
"Great atmosphere, thoroughly deserved it, was a massive win. Hard work pays off," Pickford told Sky.
While Liverpool, who had 77% possession and seven shots on target to Everton's six, won the League Cup earlier this season, the Premier League is the final trophy up for grabs for the Reds after they were eliminated from the FA Cup and Europa League.
