The Hoops captured the Scottish Premier League title, but surprisingly lost in the final of the Scottish League Cup to Killie and in the semi-final of the Scottish Cup to eventual winners Hearts.
Shiels said that Celtic had "massively underachieved" and "threw away" two cups, while claiming Kilmarnock's success made them the only rightful candidate for team of 2012.
If Shiels was intending to rile his fellow Northern Irishman ahead of Kilmarnock's visit to Parkhead in the SPL on Wednesday, then he succeeded.
Lennon has guided his side to the last 16 of the Champions League this season - where they will play Juventus next month - and they are odds-on to retain the SPL title, although they did lose their Scottish League Cup semi-final with St Mirren on Sunday.
Told about Shiels' comments, Lennon said: "It has nothing to do with him. He shouldn't be commenting on my team and my club, I don't comment on his.
"I don't know what context he is using that in (underachieving) but what is he saying about us this season?
"They won the League Cup and defended it very poorly, which no-one seems to touch on; they got knocked out in the first round.
"You can look at it in many ways but I think we have been brilliant.
"I think we probably overachieved in 2012, for the budget we were working with, the environment that we are playing in, the pressure the players are under week-in, week-out, the analysis that goes with it and the naivety and the youth that is in our squad.
"I think they have performed magnificently and I am very proud of them. You get the odd setback but in the main they have been fantastic."
Lennon's reference to the analysis of his players led him into another verbal attack, this time on former Celtic assistant manager Murdo MacLeod, who also played for the club.
In his newspaper column yesterday MacLeod, also a radio pundit, said of Celtic's defeat to St Mirren: "Blowing one treble is careless, but blowing two is verging on criminal...That's why I feel sorry for those supporters who turned up at Hampden and forked out hard-earned cash to watch that display."
Lennon said: "A columnist yesterday made a sycophantic comment about feeling sorry for the Celtic fans and their hard-earned cash.
"Personally, I don't think he cares about supporters' hard-earned cash because he will not have to pay to get into a game anywhere.
"Basically, (it is) trying to put a wedge between the players and the supporters, throwing in these cliched lines that are regurgitated every time that we get a defeat.
"I think our fans have had good value for money this season and I hope that continues for the remainder of the season.
"The columnist also described us losing the treble as careless, as if winning the treble is as easy as shelling peas.
"It is a very difficult thing to do and, as I have been reminded constantly, there are only two managers in the history of the club who have done it before.
"Some people's expectations are that we should be decimating everyone, that is not the case either, winning games is very hard to do. But I think in my time here our win percentage is excellent."
However, Lennon did admit to concerns about psychological barriers his players may have at playing at Hampden after losing to Kilmarnock, Hearts and the Buddies in their last three appearances there.
"I actually do think there might be a psychology thing about playing at Hampden for some reason," he said. "The previous two games we played pretty well but that (against St Mirren) was the poorest we have played there for a while.
"We were in scintillating form going into the game but they can't seem to replicate that at the national stadium at times so that is something I need to address if we get there again.
"It might be the atmosphere but Hampden is not there for our approval, it is not there for us to swan in and win the game and go home again.
"We have to overcome whatever mental deficiencies we have in playing there, but there is definitely a dip in performances when they go and play there."
Meanwhile, Lennon said there has been no further interest in striker Gary Hooper after Norwich had two bids rejected, but he expects to see new signing Tom Rogic in Glasgow as soon as today.
The 20-year-old midfielder signed a four-and-a-half-year contract in moving from Central Coast Mariners for about £400,000.
Lennon said: "I am hoping he will be back today or tomorrow. He has been away getting authorised by the British Embassy, he may be available for the weekend."
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