Bale firstly set up Jan Vertonghen for Spurs' opener with a lovely lofted pass in the seventh minute before lashing home his 27th goal of the season for club and country after 21 minutes.
Michu pulled one back for Swansea but Bale's brilliant block on Nathan Dyer in the final minutes ensured their fightback fell short and Tottenham climbed to third in the Barclays Premier League table to boost their Champions League aspirations.
Laudrup, one of Europe's finest midfielders during his own playing career, said: "They had a very good start but both goals showed a player of Gareth Bale's quality can make the difference. With the first we could have done better even if it was a good pass and finish. But the second goal was just a fantastic finish by a very good player."
He added: "There are different levels of players, even at this level. There are good players, very good players and exceptional players and they are the ones who sometimes out of nothing can make a difference.
"If you see his goal, or his goal against West Ham, it was not even a chance. He is a player who makes the difference and you are lucky if you have that. Not many teams do. If you take Bale out, without him the teams were at the same level."
The Welsh club suffered a third straight defeat since reaching the 40-point mark but the Dane was pleased with his side's comeback effort.
"After the two goals we reacted well and we should have scored one (through Nathan Dyer) that was harder to miss than score," he added.
"We talked about it at half-time, we felt they could be tired after having so many players on international duty and in the second half we dominated, we scored one and got close, but not close enough, to a second."