A Russian soldier was killed and another injured by shots fired at a number of Russian army checkpoints that had been the targets of at least 10 separate attacks in a 24 hour period, the administrator said on condition of anonymity.
In a separate incident, one Chechen police officer was killed and four others wounded in fighting southwest of Grozny.
Two fighters also died in the confrontation.
Two other suspected separatists were killed in the region of Vedeno, the Moscow-backed authority said.
In another clash in the Shali region south of Grozny, a separatist was killed and a Chechen police officer wounded.
Early on Saturday the body of a Chechen police officer was found in a district of Grozny.
This violence follows the announcement by officials in Grozny of five deaths and 11 injuries in fighting between separatists and Russian security forces on Friday.
Russian troops stormed Chechnya in October 1999 to try to re-establish control following defeat in a 1994-1996 war against separatist fighters.
Major clashes are now rare, but Russian forces and local Chechen allies continue to suffer casualties almost daily.