Federer pushed to the limit by canny Stepanek
The 32-year-old Grand Slam record-holder -- who has won the Dubai title a record five times -- struggled for long periods against a 35-year-old who is barely a full-timer on the ATP Tour these days.
Federer was four times within a point of going 0-3 down in the final set, any one of which could have inspired the talented and unpredictable Czech to close out the game.
Once Federer had held on in that game, and broken back, he created some momentum of his own, and the ace which took him to 4-2 signalled he had turned the tide.
"It was a difficult match and a bit typical of a second round at Dubai, and I found it difficult to get rhythm," admitted Federer.
"Radek is talented and agile, and it didn`t look good for a while in the third set.
"It was a bit frustrating. I didn`t take advantage after the first set when you hope to keep it going.
"At the start of the second he stuck in a quick break and I was under pressure for the rest of that set.
"I had to figure out how to deal with it. I am glad I made it."
Stepanek -- who has played only four games on the Tour this year -- also showed a remarkable ability to raise his game as it became evident he had a chance of pulling off a big result, and he mixed up his tactics with unorthodox brilliance.
When Federer broke back from 5-3 down in the second set he seemed to be out of danger.
However the opening point of the tie-break had to be played three times - Federer winning a Hawkeye appeal and the umpire over-ruling the next line decision - and that appeared to have an effect on the Swiss.
When Federer double faulted it left him trailing 6-2, and although he saved two set points, Stepanek was soon performing one of his notoriously extravagant celebrations and hop-skipping his way into a deciding set.
Federer turned it round by reining in his ambitions, and mixing in a little containment with his forcefulness and will take note of that when he takes on Lukas Rosol, conqueror of Rafa Nadal at Wimbledon two years ago, in Wednesday`s quarter-finals.
The last match of the day saw Novak Djokovic, who will equal Federer`s record of five titles if he succeeds again, also reach the quarter-finals, beating Roberto Bautista Agut, the world number 51 from Spain, 6-1, 6-3.
Although it was only Djokovic`s second match in more than five weeks, the top-seeded Serb was untroubled, and earned a meeting with Mikhail Youzhny, the Russian who has reached the Dubai final twice.
If Djokovic beats Youzhny he should have a semi-final with Federer.
Earlier Stepanek`s compatriot Tomas Berdych cruised into the quarter-finals and looks in the type of form that could see him reach the final for a second successive year.
The world number six was the only top ten player not to have won a title last year, though an overwhelming 6-2, 6-1 victory over Sergiy Stakhovsky now hinted how his chances of remedying that are improving.
Berdych has lost only ten games in four sets of tennis here, his steep serve and fierce forehands proving formidable weapons, and the win was his 13th in 14 matches.
He next plays Jo-Wilfried Tsonga, the fifth-seeded Frenchman, who rushed here from contesting the final in Marseille on Sunday, and who was pleased to get a day`s rest when Nikolay Davydenko, the former world number three from Russia, withdrew with an injured rib.
AFP ------------------------ ZEENEWS.com