London, May 9 : Everton and Nottingham Forest escaped the relegation zone in the Premier League after three dramatic games reshaped the relegation battle. Everton delivered their best performance of the season, defeating Brighton 5-1 away and climbing out of the bottom three in the Premier League, pushing Leeds United into the relegation zone, reports Xinhua. Sean Dyche's team had a remarkable first half, with Abdoulaye Doucoure scoring in the first and 29th minutes to put Everton 2-0 up before Jason Steele turned a low cross into his own goal, making it 3-0 after just 35 minutes. Brighton improved after the break, but Everton defended well, and Dwight McNeil netted a fourth in the 76th minute. Although Alexis MacAllister pulled a goal back three minutes later, McNeil's second of the game deep in injury time confirmed an astonishing win. Nottingham Forest won a thrilling match against Southampton 4-3, also moving out of the relegation zone. Taiwo Awoniyi put Forest 2-0 up against the bottom-ranked side before Carlos Alcaraz pulled a goal back. Morgan Gibbs-White's penalty gave the home side a 3-1 lead at the break. Vojnovic Lyanco made it 3-2, and Danilo scored another for Forest. James Ward-Prowse netted a 95th-minute penalty, adding late drama to a match that didn't finish until the 101st minute. Leicester City joined Leeds in the bottom three after a 5-3 defeat away to Fulham, where their defense allowed rivals too many opportunities. Fulham led 4-0 with goals from Willian, Carlos Vinicius, and two strikes from Tom Cairney, while Jamie Vardy missed a penalty for Leicester. Harvey Barnes pulled a goal back for Leicester before Willian made it 5-1 for Fulham. A late James Maddison penalty and a second goal from Barnes softened the result for Leicester, whose next two games are against Liverpool and Newcastle United. /IANS