Xherdan Shaqiri's first goal for Inter and a late strike from Mauro Icardi - who had earlier missed a penalty - proved enough to sink a Sampdoria side that finished the match at San Siro with nine men.
Mancini was particularly pleased with the impact of January signings Lukas Podolski and Shaqiri, who combined for the opener.
However, the Italian - who witnessed his side play out a dour stalemate at Empoli last weekend - wants to see a higher tempo and greater defensive solidarity from the team.
"Shaqiri's class and personality was clear for all to see. I'm happy that he and Podolski had such a good game," the coach told reporters.
"I think we can do even better though. Once the team is more compact they'll be able to play with even more quality.
"We need to improve our ball circulation and play high up the pitch with confidence. We also have to improve the defensive side of our game.
"Once we do that better we'll be able to push up high and risk the odd counter-attack, which is what we should have done against Empoli on Saturday."
Inter will now turn their attentions to Sunday's league clash with Torino.
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