Investigators did not say whether the three dead were soldiers or suspected drug gang members.
The military said more than a dozen guns and a grenade were seized during the operation.
"Cases such as this show the Mexican government's firm decision to continue fighting narco-traffickers," the attorney general's office said in a written statement.
Among those arrested was Ascencion Sepulveda Salto, also known as "El Gato", believed to be a powerful cartel lieutenant in Guerrero state.
Struggle for power
Edgar Luis Villegas Melendez, a Mexican general, said Vazquez had partnered with Edgar Valdez Villarreal, a US-born enforcer known as "La Barbie" in his quest for control over the Beltran Leyva cartel.
Authorities say the battle for the cartel began after Mexican marines killed Arturo Beltran Leyva, the group's leader, during a December shootout at an upscale apartment complex in Cuernavaca, south of Mexico City.
The struggle for power has triggered dozens of killings in Morelos state, where Cuernavaca is located, and in neighbouring Guerrero, authorities say.
The arrest of Vazquez and Salto "significantly affects the operations and security of the Beltran Leyva organisation", Melendez said.
The US state department said Vazquez was a key member of the Arturo Beltran Leyva cartel and that he has overseen major deals involving the trafficking of crystal methamphetamine and other drugs in Mexico, Central America, South America and the US.
He was indicted on four drug-related counts in 1997 in California.
"This significant arrest is another demonstration of the commitment and efforts being made by the Mexican government to disrupt and dismantle these violent drug trafficking organisations and bring their leaders to justice," Paul Knierim, a US drug enforcement administration spokesman, said.
At least seven major drug cartels operate in Mexico and an estimated 22,700 peoplehave been killed in Mexico's drug war since December 2006, when a military crackdown on the cartels began.