That's what happened in the original 1977 version, but George Lucas made the decision to edit the scene for the 1997 special edition re-release to make it look like Greedo shot first and Solo simply shot back in self-defense.
The Atlantic publicly uncovered the change with the article "The Star Wars George Lucas Doesn't Want You To See," in which they report "...it’s clear that Han Solo pulled out his gun and shot the bounty hunter Greedo. In the 1997 version, Greedo shoots first. In the 2004 version, they shoot at the same time."
What kind of shit is that? Lucas tells EW that the act of shooting Greedo first would essentially go against Han's core principles. "Han Solo was going to marry Leia," says Lucas, "...and you look back and say, ‘Should he be a cold-blooded killer?' Because I was thinking mythologically — should he be a cowboy, should he be John Wayne? And I said, ‘Yeah, he should be John Wayne.’ And when you’re John Wayne, you don’t shoot people [first] — you let them have the first shot."