With two games remaining for both sides, Chile moved into fifth place in the 10-team South American group on 19 points.
The top four teams in the section qualify automatically for Qatar and the fifth-placed side goes into a playoff with a team from the Asian confederation.
On a heavy pitch in the Bolivian capital, Sanchez put Chile ahead after 14 minutes but Marc Enoumba equalised eight minutes before half-time for the home side.
Marcelino Nunez restored Chile's lead after 77 minutes and Sanchez added a third with four minutes remaining.
Bolivia's top goalscorer Marcelo Martins pulled one back two minutes later but Chile held on.