Suarez and co are in real danger of missing out on the 2022 World Cup after Uruguay's fourth straight defeat in World Cup qualifiers.
2022 World Cup qualification South America
3-0
Bolivia scored twice in the first half then missed a penalty and had a man sent off in a 3-0 win over Uruguay on Tuesday, a result that revived their World Cup qualification chances and dealt a blow to their rivals' hopes of making Qatar 2022.
The defeat marked the first time Uruguay had lost four qualifiers in a row and tightens the South American race for a place in next year's finals.
Bolivia move to 15 points from 14 games, one point behind their opponents, who are level on 16 with Chile and Colombia.
The top four qualify automatically for Qatar, and the fifth-placed side go into an inter-regional playoff.
Brazil are the only country out of the 10 South American sides to have guaranteed their place at the finals so far.
In games due to be played later on Tuesday, Venezuela host Peru, Colombia welcome Paraguay and Chile host Ecuador, while Argentina entertain Brazil in the regional classico.
In La Paz, Juan Carlos Arce put Bolivia ahead after 29 minutes when goalkeeper Fernando Muslera let his speculative cross sneak through his hands into the net, and Marcelo Moreno made it two with a header seconds before halftime.
Moreno, the top scorer in the South American qualifiers, missed a penalty after 62 minutes and, although substitute Carmelo Algaranaz was sent off 12 minutes later, Arce netted his second and Bolivia's third with 11 minutes to go.
The result means Bolivia have won four of their seven home qualifiers at high altitude - a record that compares with no victories from their seven away games.
Line-ups
Bolivia XI: M. Martins, J. Arce, C. Lampe, D. Bejarano, R. Ramallo, L. Justiniano, José Sagredo, M. Villarroel, V. Abrego, R. Fernández, J. Quinteros
Subs: F. Saucedo, Jesus Sagredo, C. Algarañaz, J. Reyes, E. Saavedra, R. Cordano, B. Miranda, A. Jusino, R. Vaca, J. Gutiérrez, M. Enoumba, Franz·Gonzales Mejia
Uruguay XI: Caceres, Muslera, Godin, Vecino, Gimenez, N. Nández, Torreira, Bentancur, J. Piquerez, F. Torres, Agustín Álvarez Martínez
Subs: Suarez, D.Suárez, Campana, K. Dawson, Pereiro, F. Gorriarán, J. Rodríguez, Arambarri, G. Gonzalez, M. Ugarte, Ronald Araujo, B Rodriguez