Carvalhal said Manchester United deserved to win Saturday's game, and admitted they found it "difficult to control (Paul) Pogba and (Nemanja) Matic" in the first half.
It has been something of a mixed season for Manchester United, with the Red Devils struggling to product consistent form in all competitions.
He does at least give The Swans credit for our fightback in the second half.
Romelu Lukaku scored his 100th Premier League goal as Manchester United retook second place in the Premier League with a 2-0 victory over Swansea on Saturday, although it was Jesse Lingard who stole the show.
Last season, de Gea managed 15 clean sheets, his previous joint best alongside the 2015/16 season.
"The manager told us everybody has to fight for their place in the team", he added.
Manchester United return to Premier League action next weekend with a crunch clash against Manchester City.
"It is usual for us that we need to fight here, especially after a hard start, but it makes it more special at the end".
Palace, however, remained in trouble, only two points above the relegation zone.
"The first half was ideal", the United manager said.
"The supporters were great and they appreciated the performance of the players".
Download the Nigeria Today app from Play store. He's coming off a 2-goal, assist performance for Belgium against Saudi Arabia in an global friendly.
"In the history of football all around the world, all of the biggest clubs have phases of continuous victories and phases of transition, and these cycles go on".
Their upcoming matches, away to West Brom and at home to Everton, will likely determine whether they can cruise through to the end off the season or if they'll have to sweat it out heading into their final match against Stoke.
The Seagulls had just seen Glenn Murray's penalty saved before Vicente Iborra broke the deadlock on 83 minutes.
Snow to fall on Scotland from tonight amid Easter traffic chaos
Next week we will start off in the 50s Monday with a 50% rain chance and slowly warm back into the 60s for the rest of the week. Temperatures could drop as low as two degrees, as the region once again slips into wintry conditions.