Daler Kuzyaev fired in a long-range drive in the 27th minute and striker Aleksandr Kokorin sealed a 3-1 aggregate triumph just after the hour mark.
"We were one step away from the miracle", said Napoli's Brazilian midfielder Allan.
"Defensively, we lacked aggression, especially in the first half", Rodgers told BBC Sport after the game.
The former Start defender said: "Of course we travelled with the aim of going through and we felt during this game we could have scored a goal and that's what we needed".
Rangers edged ahead of Aberdeen into second place on goal difference on Sunday and now face a test of their staying power with today's visit of Hearts, who left Ibrox with a goalless draw in August, when the hapless Pedro Caixinha was still in charge in Govan. Ivanovic is brilliant in the air and we got blocked off.
Serie A leaders Napoli gave Leipzig a serious fright in Germany, but a 2-0 win was in vain as last week's 3-1 loss proved too much to come back from. That's about bravery and having that courage to play.
Celtic made seven changes at the weekend as they were held to a goalless draw by St Johnstone in the Scottish Premiership in preparation for the trip to Russian Federation, with only McGregor, Kouassi, Dorus de Vries and Kristoffer Ajer featuring in both games.
"Second half, we started okay, but again we conceded possession too cheaply for the third goal". "If we played with a little bit more courage and belief tonight, we could maybe have got a result".
"I think there's still an bad lot of work for us to do, but there have been a lot of positives for us in both competitions".
"I think, in the form that we're in and with the players that we have, we have options that can cause them problems whatever formation or shape they choose to play". We can learn a lot from this game but we wanted to go through also. Last week was an outstanding performance which would allow you to think you had a great opportunity to go through but if you don't start with that aggressive mentality, then it's really hard.