A close-range finish from Alba on his 50th outing for Barca in this competition killed off the match after Rafinha had opened the scoring against the side where he spent the second half of last season on loan.
PSV Eindhoven's late draw with Tottenham Hotspur earlier in the day means that the Catalans are now all but assured of a spot in the last 16 for a 17th year running, as they sit on nine points from their first three Group B matches - three better off than second-placed Inter and eight in front of PSV and Spurs.
There was little between the sides in a quiet opening quarter to the match, with Ivan Perisic - declared fit despite limping off with a thigh injury last weekend - and Philippe Coutinho exchanging wayward shots.
Handanovic had little chance of keeping out Rafinha's casual side-foot finish from Barca's next real chance, as the ex-Inter loanee justified his manager's decision to start him by converting Luis Suarez's right-sided cross.
Matias Vecino's bullet strike from range was all Inter could muster in response as the first half petered out, but the introduction of Matteo Politano at half time - surprisingly left out of the starting lineup for Antonio Candreva - saw some urgency added to their game.
Politano was involved from the off, whipping in a dangerous left-footed cross from the right that bounced awkwardly in front of Marc-Andre ter Stegen and had to be dealt with.
Some sloppy play from Arthur then gifted Politano a free shot from just inside the box, which he blazed over the bar in another let-off for the hosts.
Barca soon regained control of the match and should have had a two-goal lead before the hour, only for Handanovic to block Lenglet's latest shot with his legs, two minutes after parrying a Suarez drive over the bar.
Coutinho, who spent his formative years in European football with Inter, rattled the crossbar 20 minutes from time when Handanovic parried Suarez's downward header into the Brazilian's path.
The Nerazzurri had won their last seven matches in all competitions and could not be shaken off here, however, sending a further warning to the Spanish champions through an on-target effort from substitute Keita Balde and Lautaro Martinez's flicked finish wide of the post.
Alba did make sure of the victory when tucking the ball under Handanovic seven minutes from the end, though, rounding off a patient team move to keep Barca's European momentum going.
Handanovic had to make a ninth save of the evening late on, doing well to get behind Suarez's drilled shot to keep the scoring down to just the two goals.
BARCELONA (4-3-3): Ter Stegen; Roberto, Pique, Lenglet, Alba; Busquets, Rakitic, Arthur (Vidal 78'); Rafinha (Semedo 72'), Coutinho (Munir 88'), Suarez
INTER MILAN (4-2-3-1): Handanovic; D'Ambrosio, Skriniar, Miranda, Asamoah; Brozovic, Vecino; Candreva (Politano 46'), Valero (Martinez 63'), Perisic (Keita 76'); Icardi body check tags ::