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Gabbiadini scored twice to bring Saints level at 2-2, but had what would have been the opener incorrectly ruled out.
"Gabbiadini's first was a good goal, it should have stood. He scored three good goals," Puel said.
"It's a lot of disappointment. We deserved better," the Frenchman added.
"I would like the video for the future for this situation, but at the moment we are without. Sometimes there is a bad decision against us, it is important to accept this.
"The memories are always for the winner. It's football and congratulations for Manchester. I'm proud of my players because we played very well."
Zlatan Ibrahimovic collected the 32nd trophy of his career after heading an 87th-minute winner at Wembley on Sunday, having earlier given United the lead with a brilliant 19th-minute free-kick.
Jesse Lingard put United 2-0 in front before Gabbiadini struck twice - his fourth and fifth goals in three games since joining from Napoli for £14m.
"Puel has reason to be really sad and disappointed because he did fantastic work and his team deserved extra time,'' United manager Jose Mourinho said.
"To lose in minute 87 with very little time to try to react, as they did in the first half, is a little bit unfair on them."
Goalline technology - to determine whether the ball has crossed the line - was brought in to the Premier League for the start of the 2013-14 season.
And a trial - possibly in Britain - using video assistant referees for "game-changing decisions" will begin no later than the 2017-18 campaign, the International Football Association Board (IFAB) has said.
Southampton manager Claude Puel says he would like to see video technology used for offside after Manolo Gabbiadini's disallowed goal in the 3-2 EFL Cup final defeat by Manchester United.