Francesco Totti's second-half penalty counted for little as the San Siro side recorded a morale-boosting win and dented the Giallorossi's hopes of finishing second
After five games without a win the fallen giants have found themselves marooned in mid-table, leading to fan protests and speculation over head coach Inzaghi's future, but the gloom was lifted on Saturday.
Marco van Ginkel netted for the first time in over two years late in the first half to give Milan the lead at the break, with the hosts slightly edging a competitive opening 45 minutes.
And Roma loanee Destro doubled their lead just before the hour to lift the pressure on Inzaghi, while leaving his parent club vulnerable to being overtaken by city rivals Lazio in the race for second.
A controversial penalty - converted by substitute Francesco Totti - gave Roma hope, but Milan were able to hold on for all three points.
The defeat is a blow to Roma's hopes of achieving automatic Champions League qualifications, with Lazio able to climb to second if they can beat Inter on Sunday.
Given their recent barren spell, it was no surprise to see Milan immediately on the back foot and the hosts needed a goalline clearance from Gabriel Paletta to clear Gervinho's low cross inside the opening minute.
Eventually Inzaghi's side showed some attacking intent as Luca Antonelli met a cross-field pass in the final third, playing a perfectly cushioned header for Keisuke Honda, only for Morgan De Sanctis to make an excellent save from the Japan international.
Chances were coming at both ends and after 17 minutes Roma were denied the opener by the post, Kostas Manolas seeing his header thud against the upright with Daniele De Rossi was unable to convert the rebound.
Milan finished the half positively and should have been ahead through Giacomo Bonaventura, the forward meeting a Van Ginkel cross, but De Sanctis was equal to it with a reflex save.
They did not have to wait long for the opener, though, as three minutes later Van Ginkel ghosted in front of Radja Nainggolan at the near post to tap home Honda's low cross from the right.
The hosts doubled their lead after 59 minutes through Destro. Honda was again the creator with an enticing cross from the right that the striker headed in from close range.
Despite Milan's two-goal cushion, their nerves quickly returned when Juan Iturbe was adjudged to have been fouled by a combination of Alex and Nigel de Jong inside the penalty area.
Totti assumed responsibility from the spot and made no mistake, his penalty creeping under the body of Diego Lopez to give Roma a lifeline.
Roma continued to push for an equaliser, but Milan's defence remained resolute, Lopez making a good save low to his right to deny Iturbe and secure the points.