"It is my responsibility to say that this is as far as I can go," Mesa said on national television late on Monday, holding back tears.
"For that reason my decision is to offer my resignation from my post as president of the republic," he added, asking protesters to allow Congress to debate his successor.
"The solution to our problems must be based on the interests of all," he said.
Mesa took on the presidency 20 months ago after his predecessor, Gonzalo Sanchez de Lozada, was driven from office, also by demonstrations over control of Bolivia's gas reserves.
Congress President Hormando Vaca Diaz called for a meeting on Tuesday to elect Mesa's successor, in line with the constitution.
Diaz is next in the line of succession, followed by the speaker of the House of Deputies, Mario Cossio, and Supreme Court President Eduardo Rodriguez, who has the only legal right to call for new elections.
Because of the protests in La Paz, Vaca Diaz said he and other congressional leaders were considering alternative venues for lawmakers to meet, including Sucre or Santa Cruz.
Opposition leaders were sceptical of Mesa's announcement, recalling that on 7 March he also tendered his resignation only to have Congress reject it a few hours later.