Safin, who played with facial bruises and scratches from a recent fight in Moscow, had 14 aces on his way to a 7-6 (5), 6-4 win over Simone Bolelli in the second singles match.
Safina beat Flavia Pennetta 7-5, 6-3 to give Russia a 1-0 lead in the opener.
Safina and Safin played the Italians in a later mixed doubles match.
Safin, who is playing in the Hopman Cup for the fourth time, arrived in Perth on Saturday night with bruises and cuts on his face, including a heavy bruise under his left eye.
After Sunday's match, he first joked in an on-court television interview that the facial injuries came from "working out hard," but then said, without elaborating: "I got in trouble in Moscow. It was much worse ..."
At a later news conference, Safin admitted getting into a fight about a week ago in the Russian capital.
"It's OK, I can survive," Safin said. "Just some small problem that I wasn't in the right place at the right time. Yes, I won the fight. I'm good, I'm OK."
When Safin arrived Saturday night, he first asked for a tournament doctor.
"I wasn't sure why he was arriving so late, why he was requesting to play a day later, but now I know why," tournament director Paul McNamee said.
"I saw him in the lobby and he didn't look good. He said 'I need two things, I need to see a doctor and I need to practice.' So I got the doctor ... and he gave him some treatment and then Safin went and practiced."
The 28-year-old has slipped to No. 29 and last October contemplated retirement following his straight-sets loss at the Paris Masters. He said Sunday this will definitely be his last year on the circuit.
"My serve saved my game today," Safin said. "I couldn't ask for a better start than that."
Earlier, Pennetta was serving for the first set at 5-4 when Safina, playing her first Hopman Cup, won the next three games to take the opener. In the second, the Russian fell behind 3-0 before reeling off six straight games to take the match.
"It was not an easy match, she's a very good player," Safina said. "I think at the beginning we were both struggling with our rhythm."
Cornet beat Hsieh Su-wei 6-4, 6-4 and Simon, who moved up in the rankings from 29th to No. 7 last year, defeated Lu Yen-hsun 6-3, 7-6 (8). Cornet and Simon beat Hsieh and Lu 6-4, 7-6 (9) in mixed doubles to complete a 3-0 win.
France plays Italy on Tuesday.
Lleyton Hewitt, who hasn't played since the Beijing Olympics in August due to a hip injury and subsequent surgery, partners Casey Dellacqua when Australia takes on Germany in one of two matches Monday. Hewitt, whose ranking has slipped to No. 67, plays No. 38 Nicolas Kiefer.
The U.S, Australia, Germany and Slovakia are playing in Group A while Russia, France, Italy and Taiwan are in Group B. The winners of each group play in the final on Friday.