Darth Sidious has two things going for him: Superior power and skill, and the element of surprise.
Sidious is immensely powerful and experienced in the Dark Side. Average or even above-average Jedi are seriously outclassed. Think of Dooku vs Anakin in Ep2 or Vader vs Luke in Ep5; but in this case the difference is even greater, because the Skywalkers had exceptional raw power to compensate for their lack of experience. As Stefan's comment points out, Sidious' best tactic is to dispatch the weaker Jedi immediately to allow him to concentrate on Mace. He only has to get past their guard once to bring them down, and that's exactly what he does.
Perhaps more importantly, Mace's companions are not physically or psychologically ready for a duel with Palpatine/Sidious. Consider it from their perspective:
They have been told Palpatine is a Sith Lord, but he is able to conceal his Force powers, and he has been ruling the galaxy for the last decade. He and the Jedi have been working together on strategy in the Clone Wars and they probably have spent a lot of time with him. It's like a senior US military officer being told the President is an evil kung fu master and has to be arrested. Even if he decides to go along with it, he's going to be a little hesitant.
It's highly doubtful whether they have the legal authority to arrest the Chancellor or to take command of the republic afterwards -- effectively, they are staging a coup. They may be feeling more than a little conflicted about this.
The Sith have been no more than a legend for a thousand years. Deep down, the Jedi may simply find it hard to believe they have returned.
Lightsaber duels with another Force user are very rare, and Jedi have limited time to spend on training for them. Even if they have some duelling practice, it's one thing to train against another Jedi, and quite different to face an opponent who really wants you dead.
For all these reasons, they are not prepared for Palpatine to draw a lightsaber and try to kill them, and they pay the price. OTOH, Mace himself is a lot more powerful, and after seeing his companions die he is in no doubt whatsoever about what he has to do.
Finally, out-of-universe, this is a way for Lucas to establish that Sidious is amazingly dangerous. This guy is so scary that he can cut down three Jedi like they were mannequins. It is rather clumsy storytelling, but that's the way it goes in the prequels.