A 1-0 victory on home soil in the first leg had set up an intriguing return tie at Anfield, with many still considering the Reds favourites to progress to the final despite their deficit.
However, Southampton punched their ticket to the final thanks to Shane Long's late winner that secured a 2-0 aggregate victory.
Jose Mourinho's Manchester United hold a commanding 2-0 lead over Hull City in the other semi-final, but Puel – who felt his Saints were worthy winners over Jurgen Klopp's men - says his team will not just make up the numbers at Wembley.
"It is fantastic for all the squad and a good reward for their hard work," Puel told BBC Sport.
"It was difficult to find this opportunity to play a final at Wembley. In the two legs we deserved the win. We were fantastic in the first leg at home and we had chances in the first half [at Anfield].
"In the second half it was difficult but now we go to Wembley, not just to participate but to win this cup.
Southampton captain Steven Davis, meanwhile, was over the moon with his side's run to the final and now wants to make history.
"It is huge, we have progressed every year since I have been here but I have been waiting for a chance to get to a final," he added.
"In the first leg we should have been further ahead. We have the opportunity to go and create some history."