The argument between Danforth and Abigail can be described as two people struggling for ultimate control. Danforth is a proud man, especially of his courts and their "justice". He likes to be in control of the courts and who has their say. Abigail is the master orchestrator of many of the accusations; for quite some time she has had free reign and total power-when she writhes, people are arrested.
So, Danforth, when presented with the possiblity that Abby might be faking, questions her, and it becomes very tense. Abby, when asked if it is true that she had an affair responds indignantly with "If I must answer that, I will leave and I will not come back again!" Danforth doesn't like that, and as she prepares to storm out, demands that she stay put. Danforth brings Elizabeth in and questions her about the affair. After Elizabeth's lie, Danforth very quickly turns against Proctor, and then Abby and he work in conjunction together to discredit Mary, and eventually John Proctor. So, Abby "wins", but Danforth sure makes it easy for her by very quickly believing her "bewitchments" and not accepting the idea that Elizabeth would have lied to protect her husband's honor. He is probably grateful, because if she had been proved a fraud, it would have also proved his court, and all of his judgments, false and cruel.