Adkins' second-placed side would have opened up an unassailable eight-point lead over promotion rivals West Ham United if they had won at the Riverside on Saturday.
But, despite taking the lead through Billy Sharp's strike just 46 seconds into the match, the Saints allowed Middlesbrough to take the points thanks to Nicky Bailey's equaliser and Merouane Zemmama's late winner.
Southampton -- five points clear of West Ham -- will now be keeping a keen eye on the Hammers' trip to Leicester City on Monday, when a defeat or draw for the east London club would seal promotion for Adkins' men.
If West Ham win that would leave the Saints needing a victory over already-relegated Coventry in front of a capacity crowd at St Mary's to clinch a return to the top-flight.
"We are disappointed but we know what we've got to do," Adkins said.
"It will be bouncing at St Mary's next week and if we win we'll be promoted. We'll have 32,000 fans there cheering us on, so we move on and get ready for next week."
Reading, fresh from sealing promotion in mid-week, were held to a 2-2 draw by Crystal Palace at the Madejski Stadium but still clinched the Championship title thanks to Southampton's defeat.
Brian McDermott's side had Jason Roberts sent off on the stroke of half-time with the score at 1-1 after Wilfried Zaha's 14th-minute strike and a swift equaliser from Reading defender Kaspars Gorkss.
The Royals moved in front through Adam Le Fondre's second-half goal but were pegged back by Glenn Murray's 74th-minute leveller.
At the other end of the table, Portsmouth and Coventry City were both relegated to League One alongside Doncaster Rovers, whose fate had already been sealed.
Portsmouth were deducted 10 points for going into administration earlier this season and that punishment proved decisive as seven points separated them from safety with one game remaining.
The south-coast club -- FA Cup winners as recently as 2008 -- were beaten 2-1 at home by Derby County, but even a win would not have been enough to avoid demotion to the third tier for the first time since 1983 after relegation rivals Bristol City beat Barnsley 2-0.
Portsmouth boss Michael Appleton said: "I want to have the tools to do the job properly and I want to be able to work on a level playing field in the division we're going to be working in.
"I've been planning in the back of my mind anyway for the thought of life in League One. I think it would have been stupid of me not to really."
Coventry's last hopes of survival were ended by a 2-0 defeat against Doncaster at the Ricoh Arena.
"I am bitterly disappointed. I feel for the players because it has been really tough. We have stuck in there and kept having a go," Coventry boss Andy Thorn said.
"The club has asked me if I will stay and I have asked for assurances that they will give me some help with some good players coming in.
"I've spoken to them and said that it needs to be restructured and sometimes out of this you can come back stronger, but the player side needs addressing massively and they have assured me they will."