In a lively start to the match, Jordan Ayew struck the bar from 25 yards for the visitors before Andre Carrillo headed Watford in front on 11 minutes, BBC reports.
However, four minutes to end the game, the Ghanaian forward grabbed the equalizer with a fine finish.
However, Jordan Ayew sparked the revival with an 86th-minute equaliser and substitute Luciano Narsingh completed the thrilling comeback moments before full-time.
To convince Swansea to keep him beyond this season, Carvalhal - who joins compatriots Jose Mourinho at Manchester United and Watford's Marco Silva in managing a Premier League club - might have to save his new club from relegation.
"I said to the players before the match that if you ask 100 people who understand football, 100 people would say Swansea will be relegated", he said.
"Now 98 people will be saying it, and maybe two people are saying maybe they will not". We are still not in a good position, but the confidence is higher and belief is higher.
"It wasn't exactly what we wanted, because it is impossible in two days [since Carvalhal's appointment]", said the former Sheffield Wednesday boss. I'm not thinking just because we won one game that it is paradise, full of birds and flowers. "I promise we will try very hard to change things around".
"For the disallowed goal I didn't see a big foul, it was a normal contact in the Premier League".
Meet "Steve Jobs", the Italian clothing company
But seems like a negligence in trademarking the name of the founder has become a blessing for an Italian brother duo. They put the name on their clothing line and quickly found themselves in the middle of a lawsuit with Apple .
UP: Egypt witnesses sad Friday after Mar Mina church attack
Pope Tawadros II says that since 2013 there have been 37 sectarian attacks on Christians in Egypt, almost one incident per month. Police later cordoned off the crime scene as onlookers crowded around the church, while a forensics team combed the area.
Crystal Palace 2-3 Arsenal
However, it was not enough and the Gunners ended Palace's recent unbeaten run with a decent, if not entirely convincing win. Arsenal manager Arsene Wenger claims he is "not fearful" of losing Alexis Sanchez in the January transfer window.
Russia Accuses The US Of Supporting ISIS - Again
Mr Assad's main ally Russian Federation also said on Wednesday the main battle with Islamic State in Syria was over. The U.S. had focused its forces in Syria on destroying ISIS since the group came to prominence in 2014.
Israel strikes Gaza after rocket attack
Whilst the third fell in the Nachal Oz Kibbutz in Israeli territory close to the Gaza border, reported the Times of Israel . The shells were fired while many Israeli fighter jets, and drones, were seen flying over various parts of the Gaza Strip.