Former Manchester City boss Mancini was confirmed as Walter Mazzarri's successor during the international break - marking his return to the club he left in 2008.
Mancini won three Serie A titles and two Coppa Italia crowns in his four-your spell at the Inter helm, but he failed to get his latest tenure off to a winning start at San Siro on Sunday.
Despite applying the early pressure, Inter found themselves a goal down after 23 minutes when Jeremy Menez volleyed home superbly from 12 yards.
Inter struggled in their push for an equaliser, but eventually got back on terms shortly after the hour mark - Obi driving a low effort beyond Diego Lopez.
Stephan El Shaarawy should have done better when he hit the crossbar after 75 minutes, before Mauro Icardi lifted a half-volley just over four minutes later.
But the spoils were shared in a result that will not do either side any favours as they remain languishing in mid-table.
Mancini opted for a four-man defence, in contrast to predecessor Mazzari's preference for playing three at the back, but a more conservative approach was not apparent in the opening seconds, with Milan captain Philippe Mexes lucky to escape punishment when he appeared to pull down Icardi just outside the penalty area within the first minute.
Icardi then failed to capitalise on a misplaced pass from Sulley Muntari, brought in for Nigel de Jong as one of three Milan changes, after eight minutes - thwarted by goalkeeper Diego Lopez after stealing into the box.
It was all Inter in the early stages, and Lopez had to be alert once again shortly after denying Icardi, when Fredy Guarin drew a diving save from the Spaniard with a driven effort from 20 yards.
However, it only took one shot on target for Milan to take the lead.
El Shaarawy floated a cross into the area from the left, with Menez on hand to find the back of the net with a delightful side-footed volley from a central position.
The goal seemed to settle Milan, while Inter struggled to carve out any opportunities to level as they went in behind at the break.
Giacomo Bonaventura fired wide via a deflection seven minutes into the second half, before Icardi blasted over at the other end after Lopez's parry from a Zdravko Kuzmanovic strike resulted in a goalmouth scramble.
However, Inter were level soon after - Obi picking up a loose Cristian Zapata clearance to direct his effort low into the bottom right-hand corner.
Fernando Torres had spoken during the week of his desire to end the season as Milan's top goalscorer following a disappointing start to life in Italy, but the Chelsea loanee offered nothing in attack and was hauled off in favour of current leading scorer Keisuke Honda after 73 minutes.
El Shaarawy should have restored Milan's lead 15 minutes from time, but - after seeing Icardi squander a similar one-on-one situation in the first half - he lifted onto the ball onto crossbar, before Samir Handanovic kept out Honda's first effort on goal.
Icardi sent a half-volley agonisingly over before time was up, and Andrea Poli saw a last-gasp effort deflected wide by Andrea Ranocchia as the sides claimed a point apiece.
