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.
In mixed doubles, Pennetta and Bolelli beat Safina and Safin 5-7, 6-4 and 10-2. Any match win counts for points which would be used in tiebreakers to determine placings later in the tournament.
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 press 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 Safin 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 at the Hopman Cup for the first time, 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. "At the beginning we were both struggling with our rhythm.''
On Saturday's first day of the tournament, singles wins by Alize Cornet and Gilles Simon led France to a 3-0 win over Taiwan.
France plays Italy on Tuesday while Russia doesn't play again until Wednesday against Taiwan.
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 67, plays No. 38 Nicolas Kiefer.
The defending champion United States, represented by Meghann Shaughnessy and James Blake, takes on Slovakia in the other Monday match.
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.
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