In the first set, he sent a forehand long to lose his serve in the seventh game as Mayer before the Czech came back and won the next three games.
Mayer took a 4-2 lead in the second set before wasting a break point that would have had him serving for the set with a 5-2 lead.
Struggling with his major weapon, Berdych put only 47 percent of his first serves into play in the second set and was broken again in the ninth game before Mayer served the set out at love.
With his serve improved, Berdych took a 3-0 lead in the third set.
In his first cup singles this year and just days after his second Grand Slam doubles title at the U.S. Open on Sunday, Stepanek played in his typical attacking style, serving 13 aces and hitting 38 winners on a fast indoor hard court at O2 Arena.
"I was trying to be active," Stepanek said. "It was an important match for me. I faced a great opponent, but I wanted to use to my favor the surface and the fans."
Stepanek converted his first match point to improve to 5-2 against Monaco.
Stepanek also beat Monaco with the same set figures to clinch the 2009 Davis Cup quarterfinal for the Czechs, also at home on a fast court in Ostrava.
"It's just coincidence," he said. "But I think I was the better player in those two matches."
Seeking its first title, Argentina is without its star player Juan Martin del Potro who skipped the Davis Cup this season.
Stepanek is a key player on the Czech team but missed the first-round victory at Switzerland because of a neck operation and was able to play only the doubles in the 3-1 quarterfinal win at Kazakhstan.
He and Berdych played all five matches in last year's final in Prague and beat Spain 3-2.
Stepanek could not find his rhythm at the beginning of the first set and was frustrated at the net by Monaco's precise passing shots.
The Argentine took advantage of his first break point to go 3-1 up but allowed Stepanek to get back into the set with a double-fault on break point in the following game.
"He played so much better than me. I had my chances in the first set but couldn't make it. He impressed me," Monaco said.
Stepanek wasted a chance to serve out the first set after Monaco sent an overhead long to go 6-5 down but dominated the following tiebreaker with his volley play.
The Czech improved his game in the second, jumping 3-0 up. Monaco still fought back before double-faulting to present Stepanek a chance to serve out the set at 5-3.
"I tried to serve little bit faster," Monaco said.
The winner of the best-of-five series faces Serbia or Canada in the final in November.