Then Republicans took Congress in 1994 with their "Contract With America". They passed a budget that was much closer to balanced, whether Clinton wanted it or not. Eventually, as you say, that led to a surplus in 2000. (That was the first surplus since Andrew Jackson, by the way.) But Clinton by himself wasn't going to do it. He was going to play pretend. It was the Republican Congress that forced him to.
Now, if you think I'm advocating for the Republicans here, I note that Republicans don't believe in anything like a balanced budget when the president is their guy. I'm just saying that, if you're going to point to Clinton, state the record accurately.
(One detail I don't remember: Why did Clinton's 1993 deficit plan get no Republican votes? Was it because they did not want to cut the deficit? Was it because they just wanted to deny Clinton a victory? Or was it because they wanted to actually do something about the deficit, and Clinton's plan was more pretend than real?)
There was also the dot-com boom, which of course wouldn't have happened without Al Gore inventing the Internet.