The arrival date of humans in South America is still unclear but it appears that there may very well have been some overlap between Smilodon populator (extinct circa 10,000 years BP) and humans. From Wiki:

The 14,000 BP immigration date maximum, however, has been challenged. Claims have been made for human presence in the 20,000-30,000 BP timeframe at Pennsylvania's Meadowcroft Rock Shelter and in California's Yuha Desert as well as sites in South America [re: Chile], Central America, and Mesoamerica.

Like many other sites, there are Mastodon bones present at Monte Verde. Smilodon may very well have been over-specialized and preyed only on big animials but then so do (did) we. Loss of habitat and loss of food also lead to extinction.