The final outcome won't be known until November 22nd, but on Sunday a total of 27 senators were elected in the first round of voting, leaving 19 seats still to be decided.
The upper house, also known as the Council of States, consists of representatives from each of Switzerland's 26 cantons.
Twenty cantons elect two senators each, while six small cantons send one apiece.
Unlike in the proportional voting system for the lower house of parliament, members of the senate are elected by majority vote, leading to results that are completely different from the lower house.
The right-wing Swiss People's Party, for example, which has the largest representation in the house of representatives (or national council) has so far elected only five members — all incumbents — to the senate.
The centre-right Liberals are currently leading with eight senators, followed by the centrist Christian Democrats (seven) and the Social Democrats (six).
Second rounds of voting for the upper house are set for the cantons of Geneva, Vaud, Fribourg, Valais, Bern, Ticino, Lucerne and Obwalden.
New elections will also be held in cantons where one senator has already been elected in each, including Zurich, Saint Gallen and Aargau.
The Swiss People's Party are hoping to boost their representation in the senate from the five seats they won in 2011.
In the last election the Christian Democrats (13) and the Liberals (11) together had enough seats to form a majority in the senate.
The Socialists won 11 seats, while the rest of them were scattered among the smaller parties.
Because of the majority vote situation, some parties that may not have strong enough candidates in a particular canton are making tactical decisions to throw their support to hopefuls from similarly aligned parties.
In the canton of Geneva, for example, the right-wing parties are looking for ways from stopping the left-wing incumbents Liliane Maury Pasquier (Socialists) and Robert Cramer (Greens) from being re-elected.
Pasquier and Cramer led the first round.
The fourth-placed candidate Raymond Loretta of the Christian Democrats bowed out of the race on Monday, throwing his support behind Liberal candidate Benoît Genecand, the third placed finisher, with the hope of “breaking the monopoly of the left and allowing for a balanced representation in the canton”, the Tribune de Genève reported.
Similar tactical decisions are being made in other cantons, with parties lining up in blocs.
The parties are taking the elections seriously because, despite its fewer members, the senate has an equal amount of legislative power as the 200-member lower house.