Saints came into the match rooted to the bottom of the standings after four successive defeats and looked on course for another loss when Darren Bent gave Villa a first-half lead.
Jason Puncheon saw a shot deflected in by Villa defender Ciaran Clark shortly afterwards, before Lambert netted from the spot in stoppage time to secure a victory that helps lift the pressure off the newly-promoted side.
"I think it is understandable when you lose the first games of the season [there is a bit of tension and anxiety]," Adkins said. "It is always a situation that could be pressurised, if you like. We keep talking to players about working through the process.
"We want to pass the football and everyone is well aware of that. We've done it on many, many occasions. I think the supporters understand that as well and there was a massive togetherness all the way through the stadium together.
"One or two passes may have gone astray but there was a real belief about what we do. We scored some good goals and arguably could have got a couple more.
"It is great that we have got a football club where everyone is united together."