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The Gunners will head back to Wembley on May 30 aiming to defend the trophy after edging past the Sky Bet Championship side, who had rallied after going behind just before half-time to a fine goal from Alexis Sanchez.
Reading drew level on 54 minutes when Garath McCleary's volley was deflected past goalkeeper Wojciech Szczesny, who could only palm the ball back over the line.
Although Arsenal did twice hit the woodwork through a header from defender Gabriel, which was touched onto the crossbar by Federici at full stretch, and a close-range shot by Aaron Ramsey, the Barclays Premier League side could not find a winner in normal time.
In the end, an error from Australian Federici proved so costly, the keeper allowing a low, angled shot from Sanchez to slip through his legs and roll over the line on 115 minutes.
"It was a difficult game against a well organised, resilient, committed and physically strong Reading team," Wenger said.
"They really wanted it Reading, they were willing to die on the pitch to go to the final and you have to congratulate them for the effort they put in.
"In the end, we were a bit lucky because their keeper made a mistake and I can understand that he is distraught, but as well he kept them in the game for long periods."
Wenger added: "What was missing was a bit of speed of combination and sharpness in the final third maybe, but overall the concentration was there.
"I don't think we switched off (for the goal), but after half-time we did not come on with the same intensity. We started to suffer and we paid for that."
Wenger refused to single out Szczesny for blame on the Reading goal.
"I would have to see it again," he said.
"At first I thought it was not in because it took some time for the referee to give the goal, so I thought 'okay, we escaped'.
"But on slow motion maybe he goes back a bit to save the ball with his feet, then when the ball bounces, it was certainly over the line.
"We will have to analyse it, but the cross should not have come in and after the guy was free, so it was a combination of mistakes."
Wenger said an ankle injury to defender Per Mertesacker "does not look good", which could make the German defender a doubt for Sunday's Barclays Premier League game against leaders Chelsea at the Emirates Stadium.
Arsenal reached a record 19th FA Cup final, and will be looking to overtake Manchester United's mark of 11 victories, with five coming under Wenger's tenure.
"Now, let's focus on our next game, we come back here in the final, but we have time to prepare," Wenger said.
"The final is a 50-50 (game). If we can beat the record we will be very happy.
"Let's hope we have all of our players available and come back in good shape."
Chelsea look in a strong position to go on and win the title, having defeated United 1-0 at Stamford Bridge on Saturday evening.
Wenger said he was unaware of the final result when quizzed on its significance at the post-match press conference.
"I ignore completely what happened. Chelsea won 1-0? The usual," the Arsenal boss said. "We will see next week."
