Sanchez gave the Premier League side a 39th-minute lead from close range before Garath McCleary volleyed home an equaliser shortly after the break.
Reading needed the post and keeper Adam Federici to keep the scores level as the game required an extra 30 minutes to decide the outcome.
The Australian keeper was at fault as Sanchez's effort spilled through his grasp and rolled in and although Reading pushed for an equaliser Arsenal held on.
The Gunners will now face the winners of Sunday's other semi-final between Aston Villa and Liverpool (15:00 BST kick-off) as they look to lift a record 12th FA Cup.
Their hopes of the double continue to look unrealistic though, with Chelsea recording a potentially defining 1-0 victory over Manchester United in the Premier League.
Sanchez has 22 goals this season since his summer arrival from Barcelona and, with 36 places separating these two sides, he was the difference at a sunny Wembley.
The first half went to form with Arsenal dominating their Championship opposition and enjoying 74% possession.
Per Mertesacker had seen his early header from Mesut Ozil's corner well saved by Federici, low to his left, and at the other end Jordan Obita's fierce half-volley from 25 yards was straight at Wojciech Szczesny.
Reading forward Jamie Mackie was unlucky to have been ruled offside when clean through before Arsenal scored the opener.
Ozil unlocked the Royals' defence with a wonderful pass to Sanchez, who, after controlling and turning inside, prodded past Federici from close range.
Reading levelled nine minutes after the break with McCleary volleying home Pavel Pogrebnyak's cross from six yards. Although Szczesny got to the ball, goal-line technology confirmed it had crossed the line
Federici, who would later leave the pitch in tears, produced a wonderful one-handed save to claw out Gabriel's header as Arsenal looked to get themselves back in front against a Reading side who were growing in confidence.
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