With the London side aggrieved to have Aaron Ramsey's ninth-minute effort ruled out for a marginal offside decision, Cech was called upon to make two outstanding saves as Liverpool dominated the first half.
He admitted he was relieved by his performance after making errors in Arsenal's defeat at home to West Ham on the opening day of the English season.
"When you start with your first game at a new club and everything goes wrong you have to go back to work and bounce back," he told match broadcaster Sky Sports.
"This was one of those game where I was in the right place at the right time. It is hard to believe after all the chances in the game that it finished 0-0.
"The positive thing out of this game is the number of chances we created in the second half. We made plenty of errors in the first half."
As well as the wet, slippery conditions at the Emirates Stadium, Cech's night was made more difficult by the absence of Arsenal's first-choice central defenders Per Mertesacker and Laurent Koscielny.
But the 33-year-old made a superb block to deny Christian Benteke in the 39th minute, as the Belgium striker tried to steer in Philippe Coutinho's low cross from close range.
The Liverpool duo had also combined to leave Cech stranded in the second minute of the game, as Coutinho cracked Benteke's cutback against the bar.
But the Czech made a vital -- if slight -- intervention just before halftime when he tipped the Brazilian's curling effort onto the post.
The second half was a different story, as instead Liverpool keeper Simon Mignolet felt the heat.
Alexis Sanchez should have equalized on the hour mark, but the Chile star -- Arsenal's top scorer last season -- hit the outside of the post from a prime position.
Ten minutes later, Olivier Giroud -- the Gunners' next best marksman in 2014-15 -- slipped before he was about to shoot, and the France forward's scuffed effort from the turf was easily dealt with.
Liverpool manager Brendan Rodgers said he was "disappointed" after dropping points for the first time in 2015-16 following consecutive 1-0 victories.
The result left Liverpool on seven points along with Leicester and Manchester United, behind leader Manchester City -- which has the only 100% record.
"I thought we were outstanding, we showed everything. You heard Petr Cech getting a round of applause for the saves he made," said Rodgers, who refused to comment on reports that striker Mario Balotelli will return to AC Milan on loan.
"Overall we kept a good clean sheet but I'm disappointed we didn't win. It is about keeping really well organized. Both goalkeepers made good saves. It was a really good clean sheet against a top team."
Arsenal boss Arsene Wenger, whose team dropped points for the second time in three games, was unhappy that Ramsey's effort was disallowed.
It was the second successive game that Liverpool benefited from a doubtful decision, with Benteke's winner against Bournemouth last Monday coming after Coutinho interfered with play from an offside position.
"We scored a regular goal that was disallowed, I don't believe I can do something about it," the Frenchman said. "At the end of the day the referee canceled out a regular goal. Like it or don't like it, it is a fact.
"We had chances but we couldn't finish them off. When you play at home and don't win, people always find problems."
Arsenal next travels to Newcastle on Saturday, when Liverpool hosts West Ham.