Lennon, who was pelted with missiles by Dons fans on a spying mission, saw Anthony Stokes hand his side an early lead but Celtic failed to build on their start and went down 2-1 at Parkhead.
Fraser Forster conceded his first goal against a Scottish team since Niall McGinn netted on Aberdeen's previous visit to Celtic Park on November 23 when Russell Anderson finished brilliantly in the 38th minute.
And Pawlett struck four minutes into the second half to end Celtic's 15-match domestic winning run.
Former Celtic players Barry Robson and Willo Flood won the central midfield battle against Scott Brown and Stefan Johansen and Aberdeen were comfortable until Celtic moved centre-back Virgil van Dijk up front. But Celtic's pressure proved too little, too late.
The Championship side had failed to score in their previous five matches but Kevin Moon, Dougie Hill and Grant Anderson were all on target for Rovers, who were twice pegged back by equalising strikes from Sam Stanton and Michael Nelson.
It was another miserable afternoon for the hosts - beaten finalists in the last two years - in this competition and Terry Butcher's side were booed off at the final whistle.
Aaron Doran had put the League Cup finalists ahead at Stair Park with five minutes to go before the break but Martin Grehan levelled from the spot on the stroke of half-time and Jamie Longworth put the League One side ahead.
However, McKay struck two minutes later to secure a 2-2 draw and a fifth-round replay and maintain Caley Thistle's hopes of a cup double.
Saints were determined to lift the gloom following their semi-final defeat by Aberdeen by the same scoreline and first-half goals from Stevie May and Frazer Wright set them up for a comfortable fifth-round triumph at Station Park.
The League One side hit the frame of the goal twice but O'Halloran came off the bench to net his first goal since his January move from Bolton and on-loan Stevenage midfielder Dunne completed the scoring with a debut strike.
Despite Rovers playing in a league below Stenhousemuir, they managed to more than hold their own and book their place in the quarter-finals.
The visitors' Ross McMillan scored a bizarre own goal in the 23rd minute when the defender headed beyond helpless goalkeeper Chris Smith.
In the 66th minute, midfielder Gary Phillips put Rovers 2-0 ahead with a left-footed shot from the centre of the box for his second goal of the season.
Sean Higgins almost provided a nervy finish, but his effort came back off the crossbar in the final minute.
On the half-hour, Scott Agnew hooked the ball across goal and Nish was there to tap home from close range high into the net.
Dumbarton had the chance to double their lead in the 41st minute when they were awarded a penalty after Christopher Kane drew a foul from Scott Bain in the area.
However, the goalkeeper, who injured himself and received a yellow card in the process, made amends by smartly saving Kane's spot-kick.
Scott Linton almost doubled Dumbarton's lead in the 69th minute, but his effort hit the crossbar.