Wales led 10-0 after the first quarter of the game and looked in control of the contest, but allowed Argentina two penalties late in the first half to give the South American side momentum going into the break.
They carried that on in the second period and scored two late tries to deny Wales a third World Cup semi-final under Gatland.
"It is a missed opportunity, obviously we are incredibly disappointed," Gatland said.
"I don’t want to take away anything from Argentina. They were a tough team to put away and they kept in there.
"It could have gone either way. We probably weren’t quite at our best, but we had opportunities."
Referee Jaco Peyper had to leave the field after 16 minutes with Wales in control, and was replaced by Karl Dickson.
"It probably didn’t help with the referee getting injured," Gatland said.
"It does throw you off. We were comfortable with Jaco Peyper and the relationship we had with him.
"It’s nothing against Karl but you do a lot of analysis on referees. We hadn’t really prepared for the change."
With Wales leading 17-12 after 64 minutes Argentine Guido Petti crashed into centre Nick Tompkins at a ruck near the Wales try line, but while there was head contact, Dickson ruled the incident legal.
"It’ll be interesting to see the feedback from the panel," Gatland said.
"If he (Dickson) feels Nick has dropped his height, I need to go back and look at it. I felt it was a penalty."
Gatland confirmed he wants to stay in his role and is already thinking ahead to the 2024 Six Nations, but it would be up to Welsh Rugby if they wanted to make a change.
"This is like an Eddie Jones question, is it?" Gatland joked.
"I’m not too sure what’s in my contract. Absolutely (I want to stay). If the Union want to get rid of me, that’s up to them."
The match was also a final one in a Welsh jersey for fly-half Dan Biggar, who previously announced his retirement after the World Cup.
"He’s been a great servant for Welsh rugby. He’s been through some incredible highs and some lows. For a youngster to come through and develop, that’s been special," Gatland said.