The set went to a tie-break which Escude led 4-2 - but Henman won the next five points, the third in particularly gutsy fashion - to seal victory in 91 minutes.
Henman will now meet Seventh-ranked Russian Marat Safin, who was too good for Paradorn Srichaphan of Thailand, winning 7-5, 6-1.
Greg Rusedski failed to match his countryman Henman's progress as he lost in three sets to Frenchman Julien Boutter.
The British No2 won the first set 7-4 on a tie-break but then lost the next two sets 6-3 and 6-4 to go out.
Boutter now goes into the third round where he will face Argentinian Gaston Gaudio.
Rusedski said: "I'm very frustrated with myself. I'm extremely disappointed with the performance today.
"If I lost the match and did the right things then I have no problem, but if I lose the match and I do the wrong things then I do have a problem."
Boutter said: "Conditions are the same for both of us. It's how the guy handles things. If you handle them right, don't get frustrated, you usually end up the winner in any game.
Despite his disappointment, Rusedski vowed to improve in time for the Masters Series event at Key Biscayne next week.
"Guys who can't clear losses keep on losing, that's part of being a tennis player," he said.
"You have to wake up tomorrow and say, 'Forget about it, learn from it'. I have nothing to defend in Miami and I really have to take advantage of that week after an opportunity lost this week."
Switzerland's Michel Kratochvil, who dismissed Andre Agassi yesterday, fell at the next hurdle, losing 6-4, 6-2 to former world No1 Marcelo Rios of Chile.
The tournament's top seed, Australia's Lleyton Hewitt, survived quite a scare from Andrei Pavel to win 4-6, 6-3, 6-0.
Tenth-seed Pete Sampras dismissed Spain's Albert Costa 6-4, 6-4 and Roger Federer, seeded 12th, got past Austrian Stefan Koubek 6-4, 6-4.
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