Laidlaw's kicking proved the difference as both sides sought ranking points to improve their chances of being seeded in the leading eight countries at the draw in May for the 2019 World Cup pools.
Scotland, who face Georgia next Saturday, have the Six Nations championship to come whereas Argentina's only remaining test until the next southern hemisphere season is against England at Twickenham on Saturday.
Scotland captain Laidlaw won his duel with Pumas flyhalf Nicolas Sanchez, who was playing his 50th international, despite a 67th-minute penalty hitting the post.
Argentina breached the Scottish line 10 minutes into the second half when centre Matias Orlando chased and caught Juan Martin Hernandez’s measured cross kick, rode through a tackle and ran round to touch down under the posts.
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But Scotland hit back within four minutes to level at 13-13 when winger Sean Maitland scored in the corner from Huw Jones’s fine run and pass and scrum half Laidlaw converted from the touchline before a further exchange of penalties.
"The doubts weren't there from last week. We were disappointed from Australia but we have learned from the loss to the Wallabies," Laidlaw, whose side were beaten 23-22 by Australia last weekend, told the BBC.
"The boys were outstanding. To win at home there is no better feeling."
A tight first half ended 6-3 to Scotland after two Laidlaw penalties to one from Sanchez, who levelled with another kick soon after the restart.
Both sides had to pull out all the stops in defence in a tight tussle in which Argentina's indiscipline let them down when they would have not been undeserving of victory.
Scotland flyhalf Finn Russell tried twice to drop a goal in the last seven minutes, his first effort going wide and the second charged down to leave it to Laidlaw to settle the match.
"It's a pretty good feeling, better than last week that's for sure," Scotland coach Vern Cotter said. "I thought we improved on things we had to nail a game like that.
"I think you saw real grit and determination to work this one out and get the win, we kept pressure on them. In the first half it became a real arm wrestle."
(Writing by Rex Gowar, editing by Ed Osmond)
