Eduardo Villegas' men were expected to handle the threat of second-bottom Guabira in relatively easy fashion in front of their Estadio Hernando Siles faithful in La Paz.
But the visitors put on a brave performance, creating several promising forays forward although the first half would end goalless.
The league leaders claimed all three points when a goal-bound header from a corner was parried by the Guabira goalkeeper into the path of Gaston Mealla, who had no trouble converting from point-blank range on 72 minutes.
The Strongest are two points clear at the top of the table through eight games, having tallied 18 points with six wins, while Bolivar and Nacional Potosi - second and third respectively - slipped up.
Bolivar were held to a 0-0 draw away at bottom side Aurora, who earned just their fourth point of their campaign.
Miguel Angel Portugal's Bolivar have The Strongest covered for goal differential (plus 14 compared to plus seven) but sit two points behind them.
Nacional Potosi came from behind to salvage a point at home to Oriente Petrolero in a 1-1 stalemate.
Potosi are one point in arrears of Bolivar, with each of San Jose (fourth) and Wilstermann (fifth) snapping at their heels on 14 points.
San Jose had a 2-1 win over Universitario, with Marcelo Gomes scoring the winner on 80 minutes for the hosts.
Wilstermann remain the only unbeaten side in the league (three wins, five draws) after a 1-1 draw on the road at Sport Boys.
Blooming snapped a five-match winless streak when they got past Real Potosi 2-1.
