Barella headed home firmly at the back post in the 51st minute, rising unchallenged to get on the end of a superb cross from centre-back Alessandro Bastoni (23), who was storming up the left side and delivered a pin-point ball.
Substitute Lukaku then converted an 82nd-minute penalty after VAR referred a handball by Benfica skipper Joao Mario (30) for review by English referee Michael Oliver, who gave the spot kick after looking at the video monitor.
The two goals hand Inter a major advantage for the return leg at San Siro next Wednesday, where they will be heavily fancied to progress to an all-Italian semi-final against either neighbours AC Milan or Napoli who meet this Wednesday.
Barella’s goal exposed Benfica’s nervy defence but the penalty was unfortunate as a Denzel Dumfries (26) cross hit Joao Mario on his head and then his upright arm inside the box.
But Inter deserved their victory, conjuring up the better chances with Benfica’s Odysseas Vlachodimos (28) making important second-half saves to deny Mkhitaryan and Dumfries and avert an even heavier defeat.
“One of the best matches of this season. We deserved to win, we played very well. We have taken the first step, we hope to do everything to advance,” Mkhitaryan told reporters.
Inter, who beat Benfica’s domestic rivals Porto in the previous round with a stoic defensive performance away in the second leg, were surprisingly adventurous at the start on their return to Portugal and dominated the early skirmishes.
But it was Benfica who conjured up the first effort on target, and soon after, Joao Mario’s tempting cross flashed across the face of goal before Inter had their first effort - a powerful long-range shot by Francesco Acerbi (35) that swerved dangerously close to the Benfica goal.
The hosts might have equalised five minutes after Barella scored but Dumfries got in the way during a goalmouth melee as Benfica’s strikers looked to sweep home a cross from Joao Mario that caused chaos.
Lukaku's precise penalty then put Inter firmly in the driving seat ahead of the second leg.
