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, but fired in four goals to notch the victory.
Adkins said: "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."
"I think it is understandable when you lose the first games of the season [there is a bit of tension and anxiety]," he added. "It is always a situation that could be pressurised, if you like. We keep talking to players about working through the process. It is great that we have got a football club where everyone is united together.
"We have come a long way in a short space of time and [even] sometimes when things aren't quite going our way, everyone pulls together. Everyone has worked very, very hard from the chairman all the way down. We have got some fantastic staff, some fantastic football players and importantly the supporters, who are outstanding.
"They get behind the players and the team. They know what we are trying to achieve, what we are achieving and, together as one, we're doing it."
After going behind, Rickie Lambert drew the hosts level and then played a part in the build-up as Nathaniel Clyne tucked away a superb ball from Gaston Ramirez. 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.
Villa manager Paul Lambert said: "The first half I thought we were worthy of it but you can't defend as deep as that and expect to get a 1-0. We had a couple of chances to make it two and we were okay for 50 minutes.
"They have been down before in games so there was no need to panic. At 2-1 you are still in the game but we have moments like that.
"I never get too high when we win or too low when we lose. The lads know, they don't need me to tell them what's good and what's not. The first half we were fine but then we gave ourselves too much to do."