Scotland led by 15 points in the third Test decider after try doubles from Duhan van der Merwe and Ewan Ashman.
But the hosts fought back and Edinburgh winger Emiliano Boffelli went over after the 80-minute mark before sealing a 34-31 win by converting his own try.
Townsend told Sky Sports: “We are obviously really disappointed because we were ahead for most of the game and we had chances – chances to score a try and were held up over the line, and chances just to be a bit more efficient with our exiting in the last five minutes.
“I really feel for the players because so much of their display was full of courage and skill.
“It was our best display of the tour and would have been a famous win but the feelings just now are the opposite of that.
“The way we played away home in front of noisy crowd, against a very good side, I am so proud of that effort.
“We just need to make sure we win those games that we dominate for so long.”
Townsend refused to criticise his players for kicking for the corner instead of goal when they were four points ahead approaching the final 10 minutes.
“We got 15 ahead by kicking towards the corner in the first half,” he said.
“We backed ourselves and just didn’t get the ball over the try line. That was unfortunate but it was something we will learn as a group.”
Scotland skipper Hamish Watson admitted they had let victory slip out of their hands.
“We dominated the game, we looked good in attack but I think we were architects of our own downfall,” the flanker said.
“Every time we got ahead, and got 15 points clear, we let them back in the game, and couldn’t exit.
“We didn’t bank the points after kick-offs. When we were receiving we just couldn’t get out of our half and gifted them points.
“All credit to Argentina, they are a very good team, they were always going to stick in there, they are a proud nation and we knew they were going to come back.
“We talked about banking the points and we didn’t do that and, when we got down to five metres out on a number of occasions, our set-piece wasn’t good enough.”
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