The result leaves Southampton just one point clear of the relegation zone following a run of only one win in their last 17 games, causing Pellegrino to call the attitude of his players into question.
"When you concede in the first minute and little by little you feel that the game will be tough, every single negative impact is massive for you. I observed some players who gave up and we cannot show this, to be honest," he told reporters at his post-match press conference.
"When you are losing 2-0, you have to try, and I think we tried to be in the game, to score one, but we were never a threat for Newcastle. The spirit of the team and the attitude is everything. We have been talking about this a lot of time from the beginning.
"You can have good players, you can have a budget, but without personality and spirit, it's really difficult to compete at this level. You can lose, you can play badly, but you have to show another face on the pitch. For this reason, I would like to say sorry to our fans because we couldn't represent our club properly. This is the worst thing that we did today."
Southampton will now turn their attention to their FA Cup quarter-final with Wigan Athletic next weekend.