Debutant Cagliari coach Davide Ballardini claimed referees gave the benefit of the doubt to the bigger clubs after television replays showed Motta's goal was offside.
Teenager Philippe Coutinho, called into the team at the last minute after playmaker Wesley Sneijder was injured in the warm-up, scored Inter's second before Joaquin Larrivey pulled one back for the visitors.
Saturday's other games produced goalless stalemates as AC Milan's run of five successive wins ended at Fiorentina and a negative, time-wasting Lazio held out for a point at Napoli.
Lazio's match was a complete contrast to the corresponding fixture last season when Napoli came from two goals behind to win 4-3 with an Edinson Cavani hat-trick.
This time, Lazio made no attempt to win the game while Napoli lacked any real bite except for a lively spell early in the second half and Cavani was barely noticed.
Ezequiel Lavezzi nearly snatched a stoppage-time winner for the hosts but his volley was superbly turned away by Federico Marchetti.
Fiorentina, where Delio Rossi was making his debut as coach after replacing the sacked Sinisa Mihajlovic, were let off the hook when Milan midifelder Clarence Seedorf thought he had volleyed the opening goal in the 18th minute.
In complete contrast to the fortune enjoyed by Inter, the Dutchman's effort was disallowed for offside when replays suggested the goal should have stood.
Claudio Ranieri's Inter got their lucky break in the 54th minute when Ricardo Alvarez's free-kick was flicked by Giampaolo Pazzini to Motta, who scored from close range.
"Certain situations create doubt and in case of doubt the referee's decision goes in favour of the strongest," said Balladini, Calgiari's third coach this season.
"It's very difficult to get an offside decision in your favour at San Siro."
Ranieri, who has complained at the number of penalties awarded against his side, admitted it was a lucky break but said that his side were due one.
"They've told me it was offside, but it's our turn to enjoy some luck for a change," he said.
Inter twice hit the bar in the first half when Giampaolo Pazzini's drive was tipped onto the woodwork by Michael Agazzi who did the same to a Mauro Zarate free-kick shortly afterwards.
Half-time substitute Alvarez was also involved in the second goal on the hour, winning possession in midfield and providing the final pass to Coutinho.
The 19-year-old Brazilian cut inside his marker and scored with a low shot which crept in by the near post for his second goal since joining Inter at the start of last season from Vasco da Gama.
Inter, who started the game in 17th place, have 11 points with Cagliari on 13 having played a game more.
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