Australian Open runner-up Safin was again stretched to the limit in his first match since his Melbourne Park exploits ended in defeat five days ago as he went down 7-6, 7-6, 1-6, 4-6, 11-9.
In the season's opening Grand Slam, Safin had overcome top seed Andy Roddick and defending champion Andre Agassi in five-set tussles before caving in during the final against Roger Federer on Sunday.
After enduring a 25-hour journey back to Europe for this weekend's tie in Minsk, Safin found himself embroiled in yet another battle.
The former world number one, on a comeback trail after an injury plagued 2003, captured the first two sets on a tiebreak only to see his cushion slip as Mirnyi roared back into contention by winning the third and fourth sets.
There was little to separate the pair in the deciding set until doubles specialist Mirnyi broke the deadlock in the 20th game.
Serving to stay in the match at 10-9 down, the towering Russian's resolve finally cracked.
Earlier, newcomer Igor Andreev gave the 2002 champions a 1-0 lead after Vladimir Voltchkov retired injured in the fourth set.
In Connecticut, the U.S. took a commanding 2-0 lead over Austria after taking the opening singles rubbers in contrasting fashion.
Robby Ginepri gave the hosts the lead after staging a stunning comeback on his Davis Cup debut to beat Jurgen Melzer 6-7, 4-6, 6-4, 6-4, 6-2.
Former world number one Roddick extended the lead in emphatic style, despatching Stefan Koubek 6-4, 6-4, 6-2 in just one hour and 31 minutes.
The U.S. can wrap up the tie with victory in the doubles rubber on Saturday in which Bob and Mike Bryan take on Julian Knowle and Alexander Peya.
AUSTRALIAN RECORD
Lleyton Hewitt thrashed Swedish teenager Robin Soderling 6-4, 6-3, 6-1 to help defending champions Australia level their tie in Adelaide after Thomas Enqvist had upset Mark Philippoussis in straight sets in the first match.
Hewitt equalled Adrian Quist's long-standing Australian record for Davis Cup singles wins when he claimed his 24th career victory to leave the first-round encounter delicately poised.
"It (Quist's record) was in the back of my mind but the main priority was to try and get us back to 1-1," Hewitt said.
"I just had to come out and play my best tennis. The first set was important but I got off to a good start and it was nice to have the win in straight sets."
In Brno, Tommy Robredo had to battle for over three hours before overcoming Radek Stepanek 7-5 3-6 7-6 7-6 to pull Spain level with the Czech Republic.
Jiri Novak had spoiled Rafael Nadal's Cup debut by beating the Spanish teenager 7-6 6-3 7-6 to give the Czechs the lead.
With Spain missing the services of their top two players, world number two Juan Carlos Ferrero and Carlos Moya through injury, Nadal and Robredo were handed the task of steering last year's runners-up into the next round.
Robredo said he knew he had to produce the goods after his team lost the opening rubber in front of the partisan crowd.
"We didn't want to be down 2-0. I knew I had to battle every point. But I also knew that I had to stick to my game, I couldn't try to do things I don't normally do," he said.
The U.S., Netherlands and Argentina were the only teams to secure commanding 2-0 leads after the opening day's play.
Argentina, the only nation in the World Group playing with two top 10 players in Guillermo Coria and David Nalbandian, easily negotiated the test provided by Morocco while Netherlands dominated unheralded Canada.
