They didn't really give Sauron a second chance, though it might have turned out that way. The issue was Eonwë, at the the end of the War of Wrath had no right to judge his fellow Ainur. So he told Sauron, if he truly repented (which he seemed to, if only very briefly), he had to return to Valinor and be judged there. He feared the consequences of that judgement though and escaped, remaining quiet for quite a while.
When Thangorodrim was broken and Morgoth overthrown, Sauron put on his fair hue again and did obeisance to Eonwë, the herald of Manwë, and abjured all his evil deeds. And some hold that this was not at first falsely done, but that Sauron in truth repented, if only out of fear, being dismayed by the fall of Morgoth and the great wrath of the Lords of the West. But it was not within the power of Eonwë to pardon those of his own order, and he commanded Sauron to return to Aman and there receive the judgement of Manwë. Then Sauron was ashamed, and he was unwilling to return in humiliation and to receive from the Valar a sentence, it might be, of long servitude in proof of his good faith; for under Morgoth his power had been great. Therefore when Eonwë departed he hid himself in Middle-earth; and he fell back into evil, for the bonds Morgoth had laid upon him were very strong.
The Silmarillion - "Of the Rings of Power and the Third Age"
The Silmarillion - Of the Rings of Power and the Third Age
