"It's a big uplift, the lads are delighted after what's been a tough few weeks for everyone involved in the football club," Austin told Southampton's website. (www.southamptonfc.com)
"We knew it would come. They've been on a bit of a slippery slope but so have we, and playing at home everybody got behind us, pushing us forward.
"That's what we need. We've been struggling and everybody's had that self-doubt, but now we've won and the performance was there."
Southampton were ruthless and could have scored more and Austin, who was also denied by the foot of the post, said the team were filled with a renewed sense of belief after breaking into the top half of the table.
"Everyone involved in the football club needs to jump on the ship now and push forward," he added. "It's great for the lads moving forward to have that belief."
Victory lifted Southampton four places to 10th, but they face a different calibre of opponent in Manchester City, who are eight points clear at the top.
Saints manager Mauricio Pellegrino knows City counterpart Pep Guardiola well, having won the 1998-99 La Liga title with him while both were playing for Barcelona.
"Pep Guardiola and I won the league together at Barcelona, it was a beautiful moment in my career," Pellegrino told a news conference on Monday. "He deserves admiration for his results. Everyone can learn from him at the moment."
Pellegrino has no new injury concerns ahead of the trip to the Etihad Stadium.
(Writing by Simon Jennings in Bengaluru, editing by Ed Osmond)
