The 24 year-old defender's three goals in three games have extended Saints' unbeaten run to six matches.
Pellegrino said: “Jack's a natural, obviously.
“The passion is really important but to work every single day in training and you need passion to react well in difficult situations.
“Without the passion and character it is difficult to play at this level."
Meanwhile, Florian Gardos will be allowed to leave when his deal expires in June.
The Daily Echo says with just six months to go on his St Mary's contract, Gardos is now able to negotiate a move to potential new clubs before departing on a free this summer.
The Romania international joined Saints in the summer of 2014, moving for £6m from Steaua Bucharest.
In total, the 29-year-old has played just 18 games in all competitions, which is nearly £333,000 per appearance during his four year stay, when his £6m fee is divided up.
When added on to his reported £30,000-a-week wages, which amount to £347,000 per each of his 18 outings in the first team, it totals at around £681,000 per match.