Walter Smith's side came through the weekend SPL action with a 10-point advantage over their rivals intact and Parkhead legend Larsson believes every single member of the team will also have to play their part between now and the end of the season.
He said: "It's a lot of work and it's not only up to Robbie - it's the whole team that has to do it.
"I know they will never give up as long as there are the points to play for, that's what it takes to play for Celtic.
"You give it your all and then see at the end of the season if you manage or not.
"There are always high expectations on you there, you've got to win the league and that's every year. You have to be able to live with the pressure.
"It's fantastic news for Celtic to get a striker of his quality. From what I understand, he is a great fan of Celtic as well and that's very important. He will know how important it is for the fans to play well."
Despite being unveiled after midnight, thousands of Celtic fans flocked to Parkhead to welcome the arrival of Keane and Larsson added: "It's fantastic up there, the Celtic fans are unbelievable.
"The way they support the team and get behind the team is fantastic. I'm not surprised they turned out in numbers because he's a very good striker and he deserves it.
"They were very nice to me when I was in the city, the fans were fantastic.
"There were never any problems, from Celtic or Rangers. They always treated you with respect.
"I think that depends on how you are as a person as well and what I know of Robbie Keane as a guy is that he's very down to earth.
"That's going to be very important there because you need to behave yourself always, because the eyes are on you all the time.
"It's like a fish-bowl, really, but it was a great experience for me and I'm sure he's going to do well up there."
Keane has so far refused to comment on whether he might stay beyond the end of the season, when his loan expires.
But Larsson believes the man who will now wear the famous number seven shirt he once wore will relish life as part of the Old Firm.
Speaking on the Gabby Logan programme on BBC Radio Five Live, he said: "It's not for me to recommend whether he stays on or not.
"If he does well - which I'm absolutely sure that he will - then there are not a lot of pleasures that are better than being up there.
"I know he has played for other big teams in the past but this is one of his boyhood dreams coming to play for Celtic.
"I'm all for it that he stays up there because Celtic need all the good players they can have."
However, another former Celtic striker, John Hartson, believes any permanent move to the east end of Glasgow is unlikely.
He said: "The problem for Celtic is that they don't really push the boat out on wages.
"Robbie will be on massive money and they could probably do that for a few months if he helps them win the league or the Scottish Cup.
"Tottenham would want a massive fee for him, probably about £10million. The better he does the higher the fee goes up.
"Whether or not they could get him on loan for another season I don't know. That would be great."
Hartson scored 119 goals during his five years in Glasgow and expects Keane to have no problems finding the back of the net during his time in the SPL.
He told BBC Sport: "He'll find it easier than how he found the English Premier League, playing for Spurs, Liverpool and for his country," said the Welshman.
"Without sounding disrespectful, you do get a lot more chances. I probably wouldn't have scored the number of goals in the Premier League that I scored in Scotland.
"He's been prolific down the years. The last year or so has probably been the most indifferent time of his career because he didn't really do it at Liverpool and he hasn't been outstanding since he's gone back to Spurs."
Meanwhile, Manchester United boss Sir Alex Ferguson also believes Keane will be a worthwhile recruit for Mowbray's side.
He said: "He's a good footballer, he's experienced and I'm sure he will be a terrific addition for them."
Scottish Premier Winner 2009/10 Win Outright: Celtic 7/4