Henman won, 6-4, 6-1, in 66 minutes against the Slovak, who had knocked out the third seed, Goran Ivanisevic of Croatia, in the first round.
The British No 2 will now meet either Magnus Norman or the fifth seed, Jonas Bjorkman, Greg Rusedski's victim in the US Open semi-finals.
Henman won the first set in 39 minutes and then broke for a 2-1 lead in the second set on his fourth break point.
Kucera had a break point in the next game but a Henman service winner cleared the danger and he went 3-1 ahead.
Kucera's serving problems returned as Henman broke him again, to love, to lead 4-1. A double fault on the first of two match points at 15-40 gave the 23-year-old the match.
In his 7-6, 6-3 first-round win over Germany's Marc-Kevin Goellner, Henman unleashed a serve recorded at more than 150mph. Although called for a let, the serve set the Briton thinking that he could improve on Rusedski's mark of 143mph.
Henman said: "I have never hit a serve anything like that fast before and I appreciate these radar speed guns are not always as reliable as they might be."
The timing of Henman's serve - 151.6mph - surprised everyone, not least the Eurosport commentators, David Mercer and Frew McMillan, who wondered if the speed gun had been recalibrated.
Later, when they spoke with David Felgate, Henman's coach, Mercer and McMillan said they could not believe the serve had registered that much faster than Rusedski's record. "David said he couldn't believe it, either," Mercer said, "but if it hadn't been a let it would have counted."Reuse content