Conte's side took a while to breakdown Genoa's stubborn rearguard, but were eventually rewarded for their persistence when Romelu Lukaku produced a trademark finish just after the hour mark.
Danilo D'Ambrosio's 79th-minute header wrapped up a crucial victory for the Nerazzurri, who bounced back from their defeat to AC Milan in last weekend's Derby della Madonnina in impressive fashion.
The only negative for Inter was Lautaro's frustrated reaction to being substituted in the second half, as he was seen punching his seat in the dugout after being replaced by Andrea Pinamonti.
Conte has played down any notion of an issue with the Argentine though, insists his outburst was a result of his failure to get on the scoresheet rather than a disagreement over his early withdrawal.
"They told me that Lautaro slammed his fists on the chair when he was replaced, but I don't think he was mad at me because we have a good relationship," the Italian boss told his post-match press conference.
"Maybe something happened on the pitch with some team-mates. The players are guys who leave without any problems after the game and forget what happened. I want people who always give their best.
"Lautaro is a very good guy and everything is fine with him. Even today he put in a lot of effort and gave his contribution. A striker lives for the goal, but it is important that they also play with their team-mates."
Conte added on Inter's latest performance: "We were good, patient and we only allowed one shot. I wasn't worried before the match and I'm not enthusiastic now because for me the situations have always been positive so far. Now we have to recover our energies and prepare for the Champions League because this will be a road with many difficulties."
The Nerazzurri were held to a 2-2 draw by Borussia Monchengladbach in their Champions League group stage opener, but will have the chance to put that result behind them when they travel to Shakhtar Donetsk on Tuesday.
Inter beat the Ukrainian outfit 6-0 in last season's Europa League semi-finals, but Conte is wary of the threat they'll pose this time around having seen them beat Real Madrid 3-2 at Santiago Bernabeu last week.
"This is a completely different situation because the one in August was a mini-knockout tournament, Shakhtar still have great qualities and the semifinal with that 6-0 does not represent the value of the two teams," he said. "Shakhtar are a highly motivated team that won 3-2 against Real, who beat Barcelona [in El Clasico] today."