Goals from Badou Ndiaye and Peter Crouch cancelled out Andy King's opener as Swansea failed to achieve the mathematical miracle they needed to survive.
Reflecting on the game, Carvalhal said that Swansea's defeat was not for want of trying.
“I think we can all agree that the commitment and attitude of the team was at a maximum,” he said, quoted by Swansea's official website.
“We fought and played to the limit. We had chances but didn’t score, but we did our best – me and the players.
“I think we have done things on the limit. I am completely exhausted – I need a holiday because we pushed the players a lot.”
Carvalhal took over from Paul Clement in December and initially had a very positive impact, leading Swansea to five wins in his first nine games including victories over Liverpool and Arsenal.
But five defeats at the end of the season ultimately brought an end to their seven-year stay in the top flight, and Carvalhal will not be carrying on at the Liberty Stadium next season.
“This is a massive club – a fantastic club with great organisation,” added Carvalhal.
“The fans are superb and the city is amazing. I would really like to thank the people for the way they have welcomed me and my staff here.
“This really is a good club and they must come back to the Premier League as soon as possible.”