The Blues currently sit fifth in the table, five points behind Tottenham Hotspur and with four league games left to play this season.
Hazard was pleased to see Chelsea remain in with a shout of silverware this season, and insists that the players will not settle for anything less than a top-four league finish and the cup.
"We gave everything," the Belgium international told BBC Sport. "We played very good football, especially in the second half, and controlled the game.
"Now we go to the final and for the championship (Premier League) again, and then we can think about the final against Manchester United.
"Southampton have good players, really good, so we had to be aware every time they had the ball and when you can score the second to close the game you have to do it.
"We will enjoy the final now... we want to finish in the top four and win the FA Cup, that's the target."
Chelsea wrap up their 2017-18 league programme with games against Swansea City, Liverpool, Huddersfield Town and Newcastle United.