British number three Baker, ranked 200th in the world, beat the 45th-ranked American 6-1 6-4 in under an hour after outplaying his opponent in the first set and withstanding a service barrage in the second to get the extra break and close out the match.
Baker - who reached the second round at Queen's last week - will now face either Germany's Philipp Kohlschreiber or Go Soeda of Japan, who meet on Wednesday.
"That's my best win in terms of ranking, no doubt about that," said Baker. "My next goal is to get to 150. If I can, ranked 200, beat a guy who's in the top 50, there's no reason why I can't, a step at a time.
"I will walk on to the court whoever I play on Wednesday and I know I've just beaten a top-50 player. If you can build on that, it just snowballs."
In other men’s matches, Marcos Baghdatis showed his class to easily oust eighth seed Pablo Andujar 6-1 6-1, while Ryan Harrison, Jeremy Chardy and Carlos Berlocq also progressed.
In the women’s draw Keothavong had little joy against Slovakia’s eighth seed and former world number five Hantuchova, who won 6-2 6-3.
Much of the action was in qualifying though, and there was good news for Britain’s Laura Robson, who beat Serbia's Bojana Jovanovski 6-3 6-2 in the final qualifying round.
Robson has been given a wildcard for Wimbledon and showed good form against the world number 103, winning in just over an hour.
Another Briton, Jade Windley, lost to Hungary’s Greta Arn in their qualifier, with Arn facing Heather Watson in the first round on Wednesday.