Even though the defeat means Lazio are just five points above the drop zone with two games to go, their supporters were glad bitter city rivals AS Roma have slipped back to second spot.
Lazio coach Edy Reja was adamant his side would try to win the game but Jose Mourinho's Inter produced a fine display to move two points clear of Roma, who beat Parma 2-1 on Saturday.
Defender Walter Samuel found the breakthrough on the stroke of half time when he expertly headed in Wesley Sneijder's cross to prompt cheers of joy throughout the Stadio Olimpico.
Thiago Motta then nodded in a corner in the second half to keep Inter on course for an unprecedented Italian treble ahead of Wednesday's Cup final with Roma and the Champions League final on May 22.
"Two games remain and it will be tough but we've taken an important step," the Brazilian told Sky TV, aware his side can win their fifth straight scudetto at home to Chievo next weekend if results go their way.
The twice European champions, Serie A's most successful club, have lurched from one bad result to another this term and respite from the misery never looked likely in Sicily despite Claudio Marchisio equalising Matias Silvestre's opener.
Luigi Del Neri's side are four behind third-placed AC Milan after the Rossoneri beat Fiorentina 1-0 on Saturday.
Strikers Edinson Cavani and Fabrizio Miccoli netted as Palermo won 2-1 at Siena to confirm the home side's relegation after they were effectively condemned last weekend because of their poor goal difference.
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