The Nerazzurri won the first leg 2-0 in Romania last week, meaning that they go through 4-2 on aggregate.
Vaslui broke the deadlock at San Siro on Thursday evening on 35 minutes from the penalty spot.
Luca Castellazzi's foul on Liviu Antal saw the referee point to the spot and send off the Inter goalkeeper, with Nicolae Claudiu Stanciu slotting home from 12 yards.
Inter left it until the 76th minute to restore parity when Rodrigo Palacio found the target.
However, Fernando Varela made it 2-1 to Vaslui just three minutes later to set up a tense finale.
Ten-man Inter, though, equalised in the second minute of injury time at the San Siro through Fredy Guarin, going through to the group stage 4-2 on aggregate.