The Saints lost 2-0 to Wigan in their first home game in the Premier League for seven years, with Franco Di Santo opening the scoring just after the break and Arouna Kone bagging a second in the 90th minute.
The south-coast outfit's manager was left ruing missed chances after his side hit the crossbar twice and forced Ali Al-Habsi into a number of good saves, and says they need to stick together as they host Manchester United next before facing Arsenal at the Emirates Stadium.
"Everyone has got to stick together, we've touched on it in pre-season," Adkins told BBC Radio Solent. "There's going to be stages where you're not going to be winning consistently as we have been for the last two seasons.
"At that stage you have to make sure everyone is sticking together and believing in the principles we've got and how we want to play and pass the ball.
"There's always going to be lessons learnt, and that's the important thing, games are going to be tough, we've got to make them tough for the opposition. And when you're in the ascendancy for long spells like we were today you've got to take the chances."
"It's a relentless division. We had opportunities to score but we haven't taken them. We've got to be ruthless, the goalkeeper's made some good saves but the harsh reality is you're going to get punished at this division.
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"Wigan have gone and scored two good goals, Kelvin [Davis] has probably not had anything else to do for the rest of the game from that point of view, but that's the division that we're in."
However, the 47-year-old remained confident in his side's ability to play attractive and attacking football, and stated that Roberto Martinez's men were tough opponents.
He added: "We want to play football in a certain manner, we've created some good chances, we've hit the bar a couple of times and the goalkeeper's made some really good saves, we were on the front foot for a lot.
"We played a good side in Wigan and they've demonstrated that for the last seven or eight years that they've stayed in the Premier League. Last year they finished very, very well and this year they were excellent against Chelsea although they lost the game."
Saints were almost unbeatable at home in the Championship last season, only losing three games at St Mary's all season. Adkins acknowledged that making their home a fortress will be even harder this year, while outlining the importance of not underestimating opponents.
He continued: "The reality is we've been very good here for the last two seasons, but being back in the Premier League is going to be really challenging, there's no easy games.
"A lot of people might have turned up thinking 'we should go and beat Wigan'. You've got to have a lot of respect for the opposition, I've always said that."