Vestergaard gave the ball away on the edge of the penalty area and paid the price as the versatile Paterson, who can also play as a midfielder and full back, pounced to finish neatly past Alex McCarthy for his fifth league goal this season.
"The goal was a bit of a shank but we played very well and we deserved the win," Paterson said.
"It was a tough game - Southampton are a tough team - but we did well. It was a tough chance to take, but you have to be alive to all opportunities and thankfully it went in."
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Defeat meant Paul Sturrock is still the last Southampton manager to start his tenure at the club with a victory, guiding them to a 2-0 win against Liverpool in 2004.
"The mentality was OK. We never gave up today," Hasenhuettl said. "I have seen some good things from my team but we made too many mistakes today. We know that we have a hard and long way ahead of us.
"In the second period our performance was better and better. We had a few chances but in this best period we nearly made an own goal. It was a terrible mistake.
"There are guys in the changing room with their heads down. It's all about staying together and taking the next step in our development."
BRIGHT START
Southampton began brightly and were unlucky not to grab an early lead when Charlie Austin's flick from Matt Targett's cross went wide. Austin had another opportunity four minutes later but could not control his volleyed effort.
Cardiff responded by creating a flurry of chances and nearly went ahead after a quick counter-attack but McCarthy palmed away Josh Murphy's shot from the left.
Nathaniel Mendez-Laing, who troubled Southampton's defence throughout the opening period, saw a fizzing shot tipped over the bar on the stroke of halftime.
Southampton had a penalty appeal rejected after the restart when Stuart Armstrong was brought down but it was as close as they got with Cardiff taking control after scoring.
The home side went close again when captain Sean Morrison headed wide eight minutes from time.
A fourth win of the season -- all at home -- saw Cardiff move to 14 points. Southampton, who are winless in 12 games, are on nine points and remain in the relegation zone.
(Reporting by Shrivathsa Sridhar in Bengaluru, editing by Ed Osmond and Ken Ferris)
