The first day split leaves everything to play for in the reverse singles on Sunday.
Spain, without the injured Rafa Nadal, are bidding for a fourth Davis Cup crown in five years, while the Czechs are aiming to become the first country in 22 years to win men's and women's team titles in the same season.
"We opened the chance for Sunday. That's a good step for us," Berdych said.
The top Czech player, ranked sixth in the world, used his booming serve to good effect and drew inspiration from the vocal home support to win 6-3 3-6 6-3 6-7 6-3.
Earlier, Spain's top player David Ferrer beat Radek Stepanek 6-3 6-4 6-4 in an equally hard-fought opening match.
Berdych and Stepanek are expected to team up for Saturday's doubles match against the Spanish duo of Marcel Granollers and Marc Lopez, who are just coming off a doubles victory at the ATP Tour Finals.
World number 11 Almagro pushed Berdych all the way and won the fourth set tiebreak with one of his 21 aces but the Spaniard's brave resistance was finally broken by the 6ft 5in Czech.
"I messed it up a bit," Berdych said of the fourth set. "I gave him too much of an advantage in the tiebreak. So I was really looking forward to playing with this crowd at my back. It is an amazing feeling."
Berdych delivered 13 aces of his own and was the first to break serve in the eighth game on his way to winning the first set. But in the second, Almagro found his own serve and had a well-timed backhand winner that broke Berdych.
Berdych used his power game in the third set to force Almagro into a few costly mistakes, with the Spaniard committing 53 unforced errors in the 3 hour, 58 minute match.
The Czechs have been keen to avoid a repeat of their last clash with Spain in the final, when they were crushed 5-0 on the clay courts of Barcelona in 2009.
They are trying to win a first Davis Cup title since Czechoslovakia lifted the trophy in 1980.
Spain, meanwhile, have won five cups since 2000, and looked ready to cruise to this year's title after the opening match.
With Nadal sidelined by a knee injury, the Spanish are relying on Ferrer, 30, who just finished the best season of his career and, at fifth, is the highest ranked player in the final.
Stepanek struggled to impose himself on the Spanish number one, who fought off the Czech's mid-set rallies.
"It wasn't a stroll by any means, it cost me a lot of effort. Above all it was hard for me to finish off the points," Ferrer said.
Ferrer looked at ease on the indoor hard court the Czech team picked for the final in the anticipation that its pace would favour the home nation.
The Czechs were also hoping to benefit from the support of 13,000 fans at the O2 Arena - the same venue where the Czech women lifted the Fed Cup trophy two weeks ago.