Seeking to cap off their debut season with a second league win, Aironi headed into the break with a 7-3 lead thanks to a converted try from winger Danwel Demas.
Tito Tebaldi kicked a penalty, early in the second half, to put seven points between the sides, but the Ospreys used all their experience to reel Rowland Phillips' men back in.
Dan Biggar, the league's top scorer this season, had an off day with the boot. He kicked three penalties from seven attempts and the visitors were left frustrated by their general lack of accurracy.
But two successful strikes from Biggar after 50 and 52 minutes closed the gap to 10-9 and centre James Hook booted the regional side home with a 75th minute penalty.
Acknowledging that it had been a hard slog, Ospreys head coach Sean Holley said: "We're disappointed with our display, we're disappointed with our form since the Six Nations.
"But you have to say the introduction of the play-offs, over the course of 22 games...the table doesn't lie and in that respect we just about deserve to be in there for our efforts throughout the season.
"We missed some crucial kicks at goal and a drop goal and showed a little bit of impatience at times in our attack, we lost the ball a hell of a lot.
"But full credit goes to Aironi. They defended really well, they tackled hard and they made it extremely difficult for us to win."
The Ospreys finished a point ahead of the Scarlets in the final table, claiming fourth place which means they face another trip to Limerick to face table toppers Munster.
"We've got to go to Thomond Park now, it's the fifth time we've played Munster this season and it'll be a cracking Saturday night.
"For us the shackles are off, admittedly we felt a little bit of pressure in the last couple of weeks. But now our season restarts."
In dry and sunny conditions Biggar missed his first shot at the posts and despite some good early territory for the Welshmen, they were let down by unforced errors.
The scrum was again a strong facet of play for the Ospreys and after the Aironi front row collapsed a set piece, Biggar landed the right-sided kick for the opening points of the game.
It was a much tighter contest in the lineout where Marco Bortolami and Quintin Geldenhuys impressed. From one such lineout steal, Justin Tipuric produced turnover ball for the Ospreys who watched Biggar miss the resulting penalty.
The Ospreys needed more points to show for their territorial dominance but they were stung when Demas was unleashed for a try in the left corner, after the visitors' defence had been sucked in.
Tebaldi tagged on the conversion and both sides threatened in the final minutes of the first half. Tipuric was brought down by a covering tackle from Gareth Krause just as he looked set to score.
Then, Aironi fly-half James Marshall hacked the ball on after Tommy Bowe and Richard Fussell had coughed up possession in midfield. The try-line was coming into range as the New Zealander broke through, but his second kick was sliced and the ball went into touch close to the 22.
Biggar missed the posts with the last kick of the half, failing to punish Aironi who were pinged for going offside near the halfway line.
The Ospreys had the better of proceedings on the resumption. Bowe went close as he chased Hook's grubber kick, but a neatly-struck 45-metre penalty from Tebaldi saw the Italians make it 10-3.
Indiscipline from the Aironi forwards allowed Biggar to cancel that score out with his second penalty from in front of the posts. He followed up with a third from a more difficult range.
These kicks were becoming hugely important as neither side looked like giving a try away in the second half. Aironi conceded another penalty at scrum time, only for Biggar's attempt to miss the left hand upright.
There were some scrappy passages of play in midfield. Aironi kept things tight but a bit of variation almost paid off for them. Horacio San Martin sent up a well-judged cross-field kick and a more favourable could have seen the onrushing Demas get in for his second try.
Marshall popped up in attack once more, breaking from deep in his half and shrugging off Matthew Morgan's challenge before being tackled into touch near the Ospreys' 22.
However, a penalty concession five minutes from time proved costly when it brought 10 metres closer to the posts. Hook took on the place-kicking duties, with Biggar off the pitch, and duly held his nerve to move the Ospreys back in front.
It was tense all the way to the final whistle. Replacement Morgan missed a drop goal that would have given the visitors some extra cushion, while Aironi's late series of attacks were spoiled by a penalty for failing to release after the tackle.
|Aironi Rugby Score Card|
|Ospreys Score Card|