In the deciding mixed doubles match, the Swiss duo claimed a 6-3, 0-6, 10-5 win over Belgium's Mertens and Zizou Bergs in an hour and 17 minutes.
Bencic and Paul coasted through the first set but won a total of just 12 points in the second set as the match went into a tiebreak.
The Swiss pair did well to shake off their collapse and refocus, taking five points in a row to give themselves a 9-4 lead and then crossing the finish line when Bergs netted a return.
"It's easy to find the energy with this team. It's just great to play with Belinda. I mean, just hats off to her to play almost 3 hours and then come out again for the doubles," Paul said.
The tie was played in gruelling conditions, with the Bureau of Meteorology issuing a severe heatwave warning as temperatures hovered around 40 degrees Celsius (104 degrees Fahrenheit) and Sunday's final being moved forward.
Earlier on Saturday, Bergs needed an energy-sapping two and a half hours on court to beat Stan Wawrinka 6-3, 6-7(4), 6-3.
In his singles match against the three-time Grand Slam champion Wawrinka, Bergs took the first set with ease but was pushed hard in the second as the Swiss captain rallied to level the match.
Bergs found himself down 40-0 on Wawrinka’s serve at 4-3 in the third set, but the Belgian clawed his way back and claimed a decisive break.
In the tie's first match, Bencic kept her undefeated run at the tournament alive as she beat Elise Mertens 6-3, 4-6, 7-6(0) and put Switzerland in front.
Bencic, a former Olympic gold medallist, has won all four singles and all four mixed doubles matches during the United Cup.
“It feels like 170 kilos fell off my shoulders - I was so stressed; I really wanted to do well, and today I felt so much pressure to not let my team down," Bencic said.
"I was really focused on myself, on breathing, and I'm super happy that I stayed tough in the important moments."
On Sunday, Switzerland will face the winner of the second semi-final between defending champions the United States and two-time finalists Poland.