Raphinha's injury-time penalty gave Leeds all three points. They move up to 15th in the Premier League, five points above the relegation zone.
City dominated possession from the start against an Everton side that struggled to gain a foothold in the game and created few real chances.
Conte had won all three of his first league tests in charge of Juventus, Chelsea and Inter by an aggregate score of 10-2.
Southampton defeated Aston Villa 1-0 in the Premier League fixture at St. Mary’s stadium on Friday.
Nuno Espirito Santo's last match in charge was a 3-0 home defeat to Manchester United in the Premier League on Saturday.
Goals by Cristiano Ronaldo, Edinson Cavani, and Marcus Rashford saw the Red Devils record a comfortable win over Spurs.
Defender Aymeric Laporte was sent off right on the stroke of half-time for denying Wilfried Zaha a clear goal-scoring opportunity.
Manchester United boss Ole Gunnar Solskjaer is under immense pressure after his team's humiliating 5-0 loss to Liverpool in the Premier League fixture on Sunday.
After taking just one point from their last four Premier League games, United are thought to be seriously considering Solskjaer's future.
Mason Mount was definitely the shining star on the day as he scored a fantastic hat-trick, with two of his goals coming after 80 minutes.