Berdych, the world number six and the Czech number one, beat 11th-ranked Almagro on the hardcourt of Prague's O2 Arena 6-3, 3-6, 6-3, 6-7 (5/7), 6-3 in three hours and 58 minutes.
In the first rubber of the day, world number five David Ferrer put holders Spain 1-0 ahead after beating 37th-ranked Radek Stepanek 6-3, 6-4, 6-4 in two hours and 58 minutes.
"We are one-all, we have opened the chance for Sunday, no matter what happens tomorrow. That's a good step for us," Berdych said after the epic rubber.
"I had more faith in myself today, that I'll make it, and that helped me get through the game," he added.
In Saturday's doubles, Berdych and Stepanek will take on ATP Tour Finals winners Marcel Granollers and Marc Lopez, and Czech captain Jaroslav Navratil said both would be "100% ready."
"I'm ready to leave everything on the court, there's nothing to save energy for as this is the last event of the year," said Berdych.
"We'll do our best to win the point tomorrow, and then let Sunday decide."
In a game of fast-serving 27-year-olds, Berdych took the first set against Almagro with a single break but Almagro retaliated in the same way in the second.
A nervous exchange with the umpire and a verdict in Almagro's favour early into the third set turned the home crowd of more than 14 000 people against the Spaniard.
Berdych then capitalised on the frantic support and took the third set, but he let Almagro come back in tie-break in the fourth, only to emerge as the winner after trading breaks in the fifth.
Almagro won 21-13 on aces.
"It was a really tough match, we played really good tennis," he said.
"He played real aggressive at the end with 4-3 in the fifth set, and that's the key," he added.
In the somewhat slower first rubber, Ferrer never let Stepanek come really close, despite predicting a close game with eyes on their five-setter in the 2009 Davis Cup finals which Spain won 5-0.
"I played very well all the match. I'm very happy with my game," the 30-year-old Ferrer said.
"I was lucky because in the important moments I played better than Radek," he added.
With five double faults and first-serve percentage just over 50%, Stepanek said "the biggest difference was in the serve."
"I wanted to put him under pressure with high percentage of the first serve, coming to the net, taking the activity, changing the rhythm."
"But I played lots of points from the second serve and then his part of the game was more efficient than mine," added Stepanek, 33.
Despite concerns about a very fast Ultracushion Novacrylic court prepared by the hosts, Ferrer said he felt comfortable.
"It was very fast but I received very well and this maybe was the key," he added.
The Czech captain said he had to be happy with 1-1 after what he saw.
"We chose the surface but the Spaniards adapted themselves very well after training here," Navratil said.
"Almagro was not much worse than Tomas who won owing to his mental strength at the end of the game," he added.
Spain are missing world number four Rafael Nadal, sidelined with a knee injury.
They are looking to add a fourth crown to their three Davis Cup trophies from the last four years, after having won the cup in 2000 and 2004.
Czechoslovakia, which split peacefully into the Czech Republic and Slovakia in 1993, won the Davis Cup in 1980.
But if the Czechs win the 100th edition of the Davis Cup finals, they will clinch a rare team double, last achieved by the United States in 1990, after their women's team lifted the Fed Cup here two weeks ago. – Sapa-AFP