MUD has dubbed the Supreme Court ruling a "judicial coup" meant to strip it of its two-thirds majority in the National Assembly.
"We've come because Venezuela is celebrating. Today democracy returns to our country," Milagros Hernández, 58, a housewife who joined an opposition rally in downtown Caracas, told Reuters.
Police and National Guard troops were deployed in the area around the National Assembly to prevent violence between opposition sympathizers and Maduro supporters, who have in the past clashed amid political tensions.
MUD leaders accuse the government of undermining the incoming assembly with last-minute appointments of Supreme Court justices and by changing a central bank law to eliminate congressional control over the bank's leadership.
A portrait of late President Hugo Chávez that hung prominently in the main congressional chamber, a symbol of what critics call illegal politicization of public institutions, was removed.
Television images showed smiling opposition deputies chatting in the company of reporters, who for years have been blocked from the floor of Congress by the Socialist Party leadership.
Maduro has recognized the results of the election, but has dismissed the new assembly as "right-wing" and "bourgeois." He has accused MUD of preparing to privatize state-run companies and roll back social programs.
"Chávez leads in heaven and Nicolás leads on earth," chanted Maduro supporters who gathered in a plaza adjacent to the National Assembly.
The main item on the agenda is the election of the assembly's speaker. The opposition has tapped veteran legislator Henry Ramos for that post. On Tuesday, a group of pro-government lawmakers walked out in protest as the new congressional leadership laid out its legislative agenda.
On Monday, Ramos said Maduro should consider resigning to save Venezuela from a political crisis, echoing the call hard-liners made in 2014 when they launched street protests that resulted in dozens of deaths.
The Supreme Court granted injunctions last month that blocked four deputies — three from the opposition and one allied with the government — from taking office after losing Socialist Party candidates filed legal challenges to the results.