"This is a special win for us as we have been working very hard as a team," said Lorenzo, who won the season opening Qatar MotoGP.
"I really tried hard to establish a gap over Dani but he kept at me until finally I succeeded at around halfway and then it was race over."
Pedrosa, who has yet to win the world title having finished runner-up three times, said it was a good result given they had not had high hopes.
"It was not my day today," said the 27-year-old, who saw Lorenzo edge ahead of him in MotoGP wins with 25.
"We had loads of problems with our grip on the tyres and it was difficult keeping on the track, especially on the corners."
Crutchlow, who finished second in France a fortnight ago despite a broken leg, was more than satisfied with his performance and said he would have probably caught Pedrosa but eased up when he saw the other Honda rider Marc Marquez crash out when in second three laps from the end.
"I knew when I saw that Marc had gone out that I was assured of a podium finish and played safe," he said.
"I am pleased with this as I had an allergy this weekend and crashed yesterday, but we did well to bounce back and take third."
Rossi's fanatical fans saw their hopes of victory for their hero dashed as the 34-year-old crashed out after Bautista came in on him and both went flying.
Rossi, who has made a stuttering start to his return to Yamaha, got back on his feet to allay fears of a repeat of the broken leg he suffered at Mugello two years ago in qualifying.
Bautista said he hadn't been aware of Rossi being on the inside.
"I didn't see anything as I went into the corner," said the 28-year-old Honda Gresini rider.
"Then suddenly I felt Valentino's bike and we both fell off."
Rossi's team-mate Lorenzo had had a terrific start taking the lead ahead of pole sitter Pedrosa.
Marquez made a stunning start moving from sixth to third on the first lap and sitting on his compatriot's shoulder.
Lorenzo repulsed Pedrosa's attempts to overtake him and finally established a significant lead with 11 laps remaining as he opened up a one second, with Pedrosa struggling for grip.
As the Yamaha rider disappeared into the distance Marquez looked the better option of the two Hondas to try and claw back his lead.
Marquez, who won the Texas MotoGP earlier in the season, nipped up the inside of Pedrosa and took second with five laps to go.
But both Pedrosa and the hard charging Crutchlow found themselves assured of podium finishes when Marquez came to grief three laps from the chequered flag, his front tyre giving way and leaving the 20-year-old looking to the heavens and cursing his luck.
-AAP
