Vivas, who worked under present Chile coach Marcelo Bielsa when he was in charge of Argentina from 1998 to 2004, agreed on a 14-month, Racing said on their website.
His task is to try to prevent Racing, one of Argentina's "Big Five", from being relegated for the second time since 1983.
Vivas takes over from caretaker Juan Barbas, who took charge for four matches after Ricardo Caruso Lombardi resigned. Counting Barbas, Vivas is Racing's 20th coach since 2000.
Racing are two from bottom of the Apertura championship table with eight points from 12 matches following their first victory, 4-0 at home to Atletico Tucuman on Saturday.
They are also low in the relegation table which is based on teams' average position over three seasons - a system introduced in 1984 after Racing's relegation to try prevent big clubs dropping out of the top flight after a poor year.
World club champions in 1967, Racing have won the league title seven times but only once since 1966 when they did so under Reinaldo Merlo in 2001.
Racing had hoped to inject the team with former German captain Matthaeus's fire before he turned them down over money issues.
They said they liked the project presented by Vivas, whose club experience has also largely been as Bielsa's right hand man at Mexican clubs Atlas and America, Velez Sarsfield, with whom the pair won the Argentine title in 1998, and Espanyol in Spain.
More recently, Vivas was in charge of Argentinos Juniors in the second half of last season but resigned in May after only two wins in 15 matches, although his ground work was recognised as having been very good.
