The hosts took control with Alessandro Florenzi opening the scoring in the 13th minute and Mattia Destro adding a second 20 minutes later.
But Rodrigo Palacio's goal moments before half-time puts Inter right back in the mix going into the second leg.
The two sides played to a 1-1 draw in Serie A at the weekend and there was again little to choose between them.
Roma started much the better side and got their reward as Florenzi rose highest to head in Ivan Piris' cross.
Twenty minutes later Inter were left rocking when another Piris cross picked out Destro who made it 2-0.
But rather than fade away, Inter fought back and Destro stretched to turn in Esteban Cambiasso's cross moments before the break to ruin Zdenek Zeman's half-time team talk.
The two sides cancelled each other out in the second half, but the scoreline will leave both with cause for optimism ahead of the second leg on April 17.