The Australian Open champion survived early service breaks by the Frenchman in each set to seal the win in one hour 35 minutes on the showpiece stadium court at the Indian Wells Tennis Garden.
In bright sunshine, Djokovic came from 0-3 down in the second set and capitalised on a rising tally of unforced errors by his pressurised opponent to book his place in the last four of the elite ATP Masters 1000 event.
He will next meet either three-times champion and second seed Roger Federer or his Swiss compatriot Stanislas Wawrinka who were scheduled to play later Friday.
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"It's been the best period of my tennis career definitely the last three months, winning a grand slam, the Davis Cup and playing really well here on the court at Indian Wells," a smiling Djokovic said courtside.
"I want to maintain this level of performance by winning tomorrow."
The 23-year-old Serb claimed his second grand slam crown at the Australian Open in January and his 20th ATP title at last month's Dubai championships. He is on a career-best 18-match win streak dating back to last year's Davis Cup final.
Gasquet, who upset eighth-seeded American Andy Roddick to reach the last eight at Indian Wells, said he continually struggled with Djokovic's return of serve which he rated the best in the men's game.
'Incredible' returns
"He returns incredible, that's for sure," the 24-year-old Frenchman told reporters. "So to make some (service) points for myself is really difficult. That's why he broke me a lot of times.
"That's a big weapon for me, the serve, and I couldn't (win) points with it. For me, Djokovic has the best return. Compared to Rafa (Nadal) and Roger, he has the best return."
Gasquet broke Djokovic in the second game of the match after his opponent double-faulted but the Serb immediately recovered by breaking the Frenchman in the third, fifth and seventh games to take control.
Djokovic, champion at Indian Wells as a 20-year-old in 2008, held serve to wrap up the first set in 43 minutes when Gasquet hit a backhand service return long.
The Serb, who had dropped only six games en route to the last eight, was surprisingly broken early in the second after pushing a forehand wide and his opponent raced into a 3-0 lead.
However Djokovic again upped his level, breaking Gasquet in the fifth and seventh games before the match ended in the 10th when the Frenchman netted a forehand service return.
Spanish world number one Rafa Nadal and Argentine Juan Martin del Potro had each booked their places in the last four Thursday.
Left-hander Nadal battled past big-serving Croat Ivo Karlovic 5-7 6-1 7-6 while del Potro advanced when Spaniard Tommy Robredo pulled out before their quarter-final with a strained left abductor muscle.
