"大快人心" , whether in a sentence, or as a stand-alone expression, means the same thing. It has a more limited meaning than "everybody is happy or pleased". "大快人心" means that people are happy, or feel vindicated collectively, when a perceived injustice has been rectified.
For example, if your government changes a bad policy to one that everybody welcomes, people are happy and pleased, but it's NOT an occasion to say 大快人心. However, if the official, who had received tons of advice and warning against the bad policy, but who had used his power to insist on it now gets fired. Then people can say, 大快人心!