Schelotto, 23, upstaged Milan's maverick striker Mario Balotelli, who failed to find the target for the first time in four games since his move from Manchester City.
Balotelli was brilliantly foiled by goalkeeper Samir Handanovic twice in the first half as he faced the team where he made his name.
A Milan league derby, with Inter as the nominal home team at the San Siro, has not ended in a draw since a 2-2 in March 1999.
Cagliari scored in the fifth minute of stoppage time to beat Torino 4-3 after their opponents had two men sent off, including substitute Abou Diop after only a minute on the pitch, in a game played behind closed doors.
The goals also flowed as eighth-placed AS Roma won 3-2 at Atalanta in heavy snow while leaders Juventus enjoyed a routine 3-0 home win over Siena, which took them seven points clear of Napoli who play at Udinese on Monday.
Third-placed Milan, clearly buoyed by Wednesday's Champions League win over Barcelona, controlled the first half and could have been several goals ahead by half-time against an Inter team mired in a mid-season domestic slump.
They went ahead in the 20th minute when former Milan man Antonio Cassano lost possession in midfield and Kevin-Prince Boateng slipped a pass to Stephan El Shaarawy, who poked the ball into the roof of the net for his 16th league goal of the season.
Handanovic somehow kept Inter alive as he pushed away Balotelli's rocketing header from under the crossbar, then stopped a sliding effort from "Super Mario" from only six metres.
Inter, beaten 4-1 by Fiorentina a week ago and one point behind Milan in fifth, came to life after the break and Fredy Guarin's close-range volley was diverted onto the post by Christian Abbiati.
Argentine-born Schelotto, signed from Atalanta in the January transfer window, came off the bench in the 68th minute for his third appearance and quickly made an impact, heading in from Yuto Nagatomo's cross three minutes later.
"It's an incredible emotion to start with a goal in the derby in front of all the supporters," Schelotto told reporters.
His goal relieved some of the pressure on Inter coach Andrea Stramaccioni, who is unbeaten in three meetings with Milan.
DEFLECTED SHOT
The mid-table action in Cagliari took place behind closed doors on safety grounds following the latest episode in the long-running saga over their move to the scandal-plagued Is Arenas stadium.
Club president Massimo Cellino was arrested this month on charges of embezzlement and false representation in the rebuilding of the tiny stadium, which has three temporary stands and one permanent structure.
A Marco Sau penalty gave the Sardinians a halftime lead, but Alessio Cerci and Alen Stevanovic put the visitors 2-1 up within 10 minutes of the restart.
Daniele Conti levelled in the 75th minute, Mauricio Pinilla put Cagliari back in front with an 87th-minute
