The vote is seen as especially important for President Park Geun-hye, whose time in office has been hampered by legislative gridlock.
Saenuri hopes to win the three-fifths of seats needed before bills can be introduced and passed by parliament.
The party currently holds only a slim majority in the chamber.
Voters are casting ballots at nearly 14,000 polling stations to elect 253 of 300 lawmakers. The remaining 47 proportional representation seats are allocated to parties according to the numbers of votes they receive overall.
President Park's administration will gain significant momentum if the governing party gains a majority of seats, The Korea Times reported, enabling it to push through labour and economic reforms before her term in office expires in about 20 months' time.
With a divided opposition in South Korea, the ruling centre-right grouping is expected to continue in power. But the election will indicate the general feeling about the government as a whole.
The economy has dominated pre-election arguments, particularly plans to make it easier for employers to sack employees.
Surprisingly perhaps, to outsiders, North Korea has not been a particularly prominent issue.
The opposition has also accused the government of being heavy-handed by clamping down on dissent and protest.
Youth unemployment rose to 12.5% in February, much higher than the South Korean average rate of nearly 5%. At the same time all the main parties have promised measures to reduce poverty among the elderly.
There is speculation in the South Korean media that the polls could end the country's two-party system, as new parties challenge Saenuri and the main opposition Minju party, which in February set what appeared to be a new world record for a combined filibuster after speaking for 192 hours.
Voter turnout is estimated to be higher than in previous general elections, local pollsters told The Korea Times.
Voters in South Korea are electing a new National Assembly with the governing Saenuri party eager to strengthen its position in parliament.