Vilanova is to step up from assistant to take the reins at the Nou Camp next season after Guardiola announced on Friday he would not be continuing beyond the end of the current campaign.
After an astonishing four-year reign, during which he led Barca to 13 titles, including two Champions League triumphs, Guardiola said he needed to take time out to recharge his batteries.
"We all wanted Pep to continue," Barca's shaggy-haired defender Puyol told a news conference on Saturday.
"Once he had decided to go for the reasons he gave, which we understand and respect, the best news we could have had was that Tito Vilanova would be staying.
"It's a blow Guardiola is going, but Tito staying means the project continues. Tito knows the team better than anyone, knows a great deal about football, and understands the philosophy perfectly.
"He is the ideal person to continue the project."
Vilanova, a former Barca youth team player, has been assistant to his close personal friend Guardiola for the last five years, starting out with him in the dugout with the Barcelona B team.
Spanish international Puyol thanked Guardiola for what he achieved with the club and admitted no one in the squad had any idea he was not going to renew his contract.
"We hadn't picked it up at all, a week ago I would have bet everything on Pep continuing," Puyol said.
"We have to thank Pep for everything he has given us. He marks a before and an after in football."
In the last week, Barca were knocked out of the Champions League by Chelsea and saw their chances of winning a fourth consecutive La Liga crown all but disappear when they lost 2-1 at home to leaders Real Madrid last weekend.
They travel to Madrid to play Rayo Vallecano in La Liga on Sunday, seven points behind Real, who host Sevilla earlier that day, with four matches left to play.
However, they still have the King's Cup final against Athletic Bilbao, who they beat to lift the trophy in 2009, which offers the chance for a 14th title of the Guardiola era before he leaves.
"You are always motivated to win a title," Puyol added, "but now what we want to do, is to win it for him."
The final is to be played at the Calderon in Madrid on May 25.