Poland international Boruc sustained a fractured hand when attempting to keep out a shot from Chelsea striker Demba Ba at Stamford Bridge on Sunday.
Ba's effort hit the post, but the ball ricocheted nicely for Gary Cahill to force the ball over the line to equalise at the start of the second half.
Gazzaniga was beaten twice more as the Blues rallied to what was in the end a comfortable victory to move back within four points of leaders Arsenal and inflict a second straight defeat on Saints.
Given the hectic Festive schedule, Boruc could miss some eight matches if he needed a six-week recovery time.
Reserve Kelvin Davis is also sidelined by a back problem, so 21-year-old Argentine Gazzaniga - signed from Gillingham and who made 11 appearances last season - looks set for an extended run of unexpected first-team action and will start against Aston Villa at St Mary's on Wednesday night.
"Of course we are going to miss Artur, he is a great goalkeeper, but we have complete trust in the whole squad, not just Paulo Gazzaniga but the whole squad," Pochettino said on Saints Player.
"It will depend on his convalescing and recuperation for how long he is out for and we do not know exactly how long that will be."
Until veteran campaigner Davis recovers, 20-year-old American Cody Cropper will offer support from the Saints development squad.
Pochettino added: "Paulo has more experience than Cropper in the Premier League, he has been part of the senior squad for a long time.
"Cody has been around the first-team squad as well, but it is clear he is still a bit underneath Paulo in terms of experience and that is why we are going to be relying more on Paulo."
Pochettino warned his players they must not let successive Barclays Premier League defeats to Arsenal and Chelsea derail what has been a promising campaign after slipping out of the top four.
However, Pochettino believes Southampton can take confidence from their performances at the Emirates Stadium - where a blunder by Boruc gifted the Gunners their opening goal - and then Stamford Bridge on Sunday, when the visitors had gone ahead after just 13 seconds through Jay Rodriguez before capitulating following injury to their goalkeeper.
"All we can do is recover, to get our energy back and try to fix the mistakes we have made in the past to make sure they do not happen again," Pochettino said.
"We may have suffered two defeats, but overall we competed very well against two very tough sides.
"We did make some mistakes which meant it was not possible for us to win those games, but in none of those cases were Arsenal or Chelsea superior to Southampton ."
Southampton have yet to lose at home in the Premier League, while Aston Villa have suffered only one away defeat, with two wins and three draws.
"Aston Villa are a strong side and very good on the counter-attack, especially when they are playing away," he said.
"Overall we are expecting a really tough matches because of all the global aspects of this match."