The Italian giants hit back impressively on Thursday night after we looked in control of the tie at 3-0 up from last week's first leg.
Goals from Antonio Cassano, Rodrigo Palacio and William Gallas' own goal levelled the aggregate scoreline at 3-3 at the San Siro.
Palacio had a chance to win it in added time but we responded well in extra time and forced a couple of fine saves from Samir Handanovic before finally grabbing the vital away goal through Emmanuel Adebayor in the 96th minute.
That meant Inter needed two more goals to go through - they scored one to level the tie at 4-4 in the 110th minute, but the fifth was beyond them as we got over the line on away goals.
"It was a difficult night for us, one of those nights where we just needed a little magic," said Lewis, who stepped into the action for Jermain Defoe in the 56th minute.
"We knew we should have controlled the match better and we didn’t play our game but we're delighted to get through to the quarter-finals, that was our aim.
"Now we can all breathe a little easier and we’re all looking towards the next game."
As always, the next challenge is never too far away and we take on Fulham back in Premier League action at the Lane on Sunday.
"It's another huge game, another derby and we’re all determined to win before we go away for the international break," added Lewis.