Based on final vote counts in half of Switzerland's 26 cantons, and partial counts in 11 others, the ATS news agency projected the party would take 11 additional seats in parliament with about 29.5 percent of the vote.
That would give it 64 of the 200 seats in the lower house, beating its previous record high of 62 seats after the 2007 election.
Along with advances made by the centre-right Liberal Party, Switzerland's third largest party, SVP's gains should tip the scale in parliament from the centre-left towards a centre-right majority.
The Socialists, the country's second biggest party, appeared set to lose three of their 46 seats, while the Liberals were poised to boost their representation to 33 from 30 MPs.
The Christian Democrats, meanwhile, were expected to lose one of their 29 seats.
The biggest loser, according to projections from the Swiss Broadcasting Corporation, were the Green Liberals, on track to lose six of their 12 MPs, and the Greens (ten seats, down from 15).
Voter turnout was 48.4 percent, slightly lower than the 48.5 percent registered in 2011.
Among the parliamentarians already sure to take a seat was Magdalena Martullo-Blocher in the canton of Graubünden.
She is the daughter of Switzerland's perhaps most controversial politician Christoph Blocher, a SVP vice president who served in government from 2004-2007 before being pushed out over his extreme positions and confrontational style.
The expected shift comes as surging numbers of migrants and refugees moving through Europe have heightened the focus on the issue in Switzerland, even though the wealthy Alpine nation is yet to be significantly affected by the crisis.
"One theme has unfortunately been very dominant during the campaign," Rebecca Ruiz, a candidate for the Socialists, told the RTS broadcaster, lamenting that "people voted out of fear."
SVP members also credited the party's harsh stance against closer ties with the neighbouring European Union for its advance.
Relations with the EU, Switzerland's top trading partner, have been badly hit by a narrow Swiss popular vote in February 2014, championed by the SVP, in favour of restricting immigration from the bloc.
The party is calling for a strict enforcement of the immigration cap and renegotiation of other bilateral agreements with the EU to ensure Switzerland remains "independent".
Second government seat?
The centre-right Liberals are expected to cooperate closely with SVP on economic and energy issues but take a very different stance on other subjects, including immigration.
It remains unclear however what impact the shift in the lower house will have.
The makeup of the upper house, the 46-member council of states (senate), remained undecided with elections in the cantons of Bern, Fribourg, Geneva, Vaud, Valais and Zurich set to go to a second round.
The balance of power in the senate, where the SVP currently have five seats, was not expected to shift dramatically.
In any case, power-sharing and consensus rule are the norm in Switzerland and the power balance in parliament is not directly reflected in government.
Once the two houses of the new parliament are in place, the joint assembly will in December elect the government, with the seven posts traditionally shared among the major parties from right to left.
Despite being the country's largest party, SVP holds just one seat.
It has its eye set on a second seat, which could be easier with a centre-right majority parliament.
Voter turnout, which has not passed 50 percent in Swiss legislative elections since 1975, was expected to tick in at just 47.5 percent, according to early results.