Technically I guess Spacewar! was the one who started e-sports, was the first game people competed in. Personally, growing up in Sweden, I think FPS (namely CS1.5/1.6) was the first game that enabled people to play games professionally on a international level, so I'll always associate CS with starting that, but again, technically I guess Quake was the first FPS people competed in professionally, at least in the US.

But it certainly was the game that made it popular across the world.

It started modern esports. There were gaming competitions in the 80s, but there weren't team houses, coaches, analysts, big money sponsors, regular huge events, dedicated TV channels, players in prime time commercials and dating actresses and pop stars, etc... Brood War hit in Korea like nothing before or after it. There were literally three full time, 24/7 TV channels showing Starcraft content at it's peak. No other game has ever done that.

In the RTS niche, it is definitely the game that started e-sports that had any sort of weight and global audience.

I'm honestly not even sure which other RTS game would be close? Age of Empires 1? I don't think it ever had the same traction or hype until AOE 2.