The Saints should have found the breakthrough themselves in the first-half, Jay Rodriguez spurning a good chance inside the area after skipping past the challenge of Ashley Williams.
And the south coast club were indebted to goalkeeper Fraser Forster for denying Romelu Lukaku several times as the home side threatened the game's opening goal.
But second-half substitute Enner Valencia fired his first goal in over a year to give Everton the lead, before Leighton Baines' penalty followed from Maya Yoshida's foul on Valencia and a late Lukaku effort gave the Toffees a kinder score-line than perhaps their performance merited.
We're in a difficult moment, admits Puel
And Saints boss Puel told journalists in his post-match press conference: "It's a difficult moment. I think it was a close game.
"I think that other teams have been clinical against us, like in the two last games. This is difficult. Today, because I think that my players stayed with a good spirit and strong character.
"We had chances before them and it's difficult to accept this, but it's a difficult moment.
It's too much of course to take three goals today after this game. It was the same feeling, for example, against Tottenham Hotspur. After 20 minutes, it was not possible to have the idea that we would concede four goals.
"Today, three goals, it's not normal because it was a courageous game. It's very difficult to accept. I think we are so unlucky for this moment.
"We lost Cedric Soares after five minutes of play. It was not possible for Ryan Bertrand to play because of his heel.
"[Sam] McQueen, to play two games in two days, for a young player, it's very difficult. But despite this, he gave a good answer today. He played 70 minutes - it was courageous."
Puel: Playing three games in five-and-a-half days is not possible
On whether he thought his side's last half-an-hour was a result of having to play three Premier League games in the last six days, all of which have ended in defeat in their worst run of league results in over two years, the Frenchman added: "We started the season with many games.
"We played at the beginning of the season, every three days, every two days sometimes, with international players also playing over the international break.
"They made very good work today. Since Boxing Day, we are the only team with three games in five days-and-a-half.
"This is very difficult and not possible. We tried to give a good answer and of course, when we are so unlucky, it's difficult to give a good answer in this moment and this situation.
"Now it's important to think positive and prepare for our cup games. They are important for us.
"But also for the other Premier League games because we're only half-way through the season. It's important to look where we are in the table and in our head also."
"They can lose confidence sometimes also, because when we are in a difficult situation, it's already difficult to score or to be clear.
"Of course it's always difficult but it's a good experience for this young squad. It's important now to recover from it and protect them. They can do work. It's important to prepare for the next game with confidence and with strong character."