Final results released Monday from nationwide balloting the previous day showed the Socialists and their allies on the left winning at least 46 percent of the vote. The main conservative bloc - led by former President Nicolas Sarkozy's UMP party - had at least 34 percent.
The unaligned far-right National Front won 13.6 percent, remaining a wild card for Sunday's second round. Tiny parties drew the rest.
Voters "have clearly expressed their desire to give Francois Hollande the means to act during this five-year term," Socialist boss Martine Aubry told reporters. "Everything has to be done, that's why this second round is absolutely crucial - and the marching order is simple: unity, and mobilization."
Hollande didn't address the election result in brief comments to reporters Monday after a meeting with Niger President Mahamadou Issoufou.
Polling agencies that have calculated the numbers precinct-by-precinct say the Socialists and allies could win an absolute majority in the 577-seat Assembly in the final round. It would cap the left's near-universal control of France's political landscape: The Socialists won control of the Senate last year; they or their allies run 21 of mainland France's 22 regional governments; Socialist mayors run many big cities including Paris, Lille and Lyon.
The Socialists were once badly riven by personal infighting and ideological clashes over issues like the ill-fated European Union Constitution referendum in 2005. But over the past year, their political-campaign management has improved, allowing the Socialists to capitalize on sluggish economic growth, high state debt and disillusionment with Sarkozy's brash style to oust conservatives from years in power.