A first-half own goal put the Blues ahead at the Liberty Stadium only for the hosts to pull level through a Jordan Ayew strike after the interval.
However, Carvalhal had Lukasz Fabianski to thank for keeping his team in the game.
An example of that was their 1-1 draw at West Brom.
"There are teams fighting at the bottom and Swansea showed a lot spirit".
An uninspiring match produced a lifeless scoreline although the Toffees nearly snatched victory at the last.
'But we would still like to have performed better'.
'It was one of our best performances of the season, but in football you can't get everything'.
Jason Roy pummels DD to a last ball victory against MI
Ishan Kishan came in at No.3 and impressed with his batting with a quick 44 in 23 deliveries. The two opening loses made Mumbai tweak their playing XI and also their batting order.
Everton were toothless in attack, with Yannick Bolasie one of their biggest culprits as his influence quickly faded out of the game.
But a better case can be made for this being a low-scoring game.
Everton will also come into the game in a confident mood after holding Liverpool to a goalless draw at Goodison Park last weekend and they could have even went on to win the game.
The swans will be without Fer and Wilfried Bony, who are ruled out to the end of the season with injuries. There are doubts over the fitness of both Mike van der Hoorn (shoulder) and Sam Clucas (knock) for this match which could be a major blow for Carvalhal, however, both are believed to be ready for selection in time for this match.
And he revealed his determination to make an impression during his return to action. That was Jordan Ayew, who is 5/1 for first goalscorer honours.
Ayew has missed Swansea City's last three matches after his red card at Huddersfield last month.
The aforementioned Wayne Rooney is 6/1 and has a good record against Swansea - four goals in his last nine appearances against the Swans - but that's no value if he carries on playing as deep as he has done recently.