Visitors threatened upset after early D'Agostino goal, but second-half onslaught from Stramaccioni's men eventually claimed the spoils
The visitors got off to a flyer on six minutes when midfielder D'Agostino pounced on a rebound to fire home, after striker Mattia Destro's strike had been palmed out by Inter keeper Julio Cesar.
Mannini fired warning that Siena weren't content to sit on their lead, however, smacking an effort narrowly wide of Julio Cesar's goal on 15 minutes.
Inter were doing their damnedest to get back on level terms, though, and Milito shaved the post with a header from a Cristian Chivu cross, before hitting the net with a header - only to see the goal ruled out for a push on Siena stopper Gianluca Pegolo.
Milito was at the sharp end of everything as the hosts piled on the pressure, but the striker's finishing let him down badly soon after when a lovely move between Joel Obi and Chivu ended with the Argentine rushing his shot and passing a tame effort straight at Pegolo.
The pressure was mounting on Siena, but Stankovic was unable to make it pay - the midfielder horribly shanking a volley wide of the target after seizing on a knockdown from Milito.
Inter's pressing paid off five minutes before the break, when Milito made good on his first-half goal threat. The Argentine levelled the scores with a little help from the visitors, however, Mannini shooting his side in the foot with a horrible attempted clearance that landed neatly on Milito's head six yards out.
The second half started nearly as brightly as the first - Siena looking to make inroads, but Inter looking the more dangerous, particularly Mauro Zarate down the left.
Milito nearly doubled his tally on 53 minutes, narrowly shooting wide under pressure after Pegolo pushed out Ricardo Alvarez's fierce left-footed strike. A flurry of attacks from the hosts then went unrewarded; Alvarez again, Zarate, Lucio and Cambiasso all letting presentable opportunities slip by.
The kitchen sink was close to being thrown at Siena at this stage, Zarate slicing through Siena's right flank almost at will - one highlight being an audacious shoulder-high take from a ball that looked certain to go out of play.
After withstanding Inter's barrage, Siena had a few chances of their own, Pablo Gonzalez shooting straight at Cesar before Destro was unable to finish off some deft footwork and blasted over.
As the match entered the last 10 minutes it was now Siena asking questions - but it was Inter who took the lead from the spot. Mannini again offered unwitting assistance, harshly adjudged to have pulled down sub Nagatomo. Milito banished memories of missing his last two penalties to blast home his second.
Stankovic went close to grabbing a third for Inter, but the hosts were nearly made to pay when Gonzalez failed to get on the end of a last-gasp opportunity to gain the visitors a draw.
However, Inter held on, and their outside chance of finishing in the top three remains intact courtesy of this triumph and Lazio's loss in Turin against Juventus.