City had the chance to extend their advantage inside the opening two minutes, but Mark Hudson managed to clear Marvin Elliott's header off the line.
The home side were well on top in the early stages, with Trundle heading straight at the keeper and Nick Carle seeing a dangerous effort deflected over the bar.
Scott Sinclair had a half-hearted shout for a penalty waved away by referee Howard Webb as the Eagles started to gain a foothold in the game.
But the opening goal came courtesy of a breakdown in communication between Robins keeper Adriano Basso and his centre-back Jamie McCombe.
Basso came well off his line to deal with Danny Butterfield's hopeful ball into the box but McCombe tried to head it clear, leaving his keeper in no man's land as Watson directed a superb looping header into the unguarded net.
McCombe went close to making amends at the other end soon after, but his header just cleared the crossbar.
A reckless tackle on Louis Carey by Palace defender Clint Hill sparked a confrontation on the touchline which resulted in Robins assistant coach Keith Millen being sent to the stands.
Shaun Derry battles for possession with Marvin Elliott
Palace ended the first half on the attack, but it was Bristol City who found the momentum in the minutes after the restart without really testing Palace keeper Julian Speroni.
The pivotal moment came on 72 minutes when Carle was harshly adjudged to have fouled Tom Soares in the box.
Watson, who had netted from the spot in the first leg, lined up his crucial penalty only to see his effort rebound off the post.
A nervy close to the 90 minutes saw the Eagles gain the upper hand against a home side sitting dangerously deep.
Interview: Bristol City midfielder Michael McIndoe
But Trundle came agonisingly close to settling it in normal time only to see his chip bounce off the bar.
A cagey opening spell of extra-time finally saw a breakthrough come with a moment of real quality from Trundle.
The former Swansea star collected a knockdown from Dele Adebola, trapped the ball and sent a delightful curling effort into the top corner.
Palace had no option but to commit players forward, and it was no surprise when a powerful free-kick from McIndoe flew in on 110 minutes to end the Eagles' hopes.
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