City battled hard at Bournemouth but were ultimately undone by an impressive Premier League side.
The Cherries, who finished sixth in the top flight last season, progressed to the fourth round thanks to a Ryan Christie hat-trick and a strike from Rayan.
City head coaches Chris Cohen and Tom Shaw made 10 changes with Mikkel Ladefoged, Chieo Ogbene and Daniel Oyegoke all making their first starts after joining late in the transfer window.
Bournemouth boss Marco Rose also switched his side up, with eight alterations from their last outing.
The Imps had an opportunity after five minutes when Callum Elder was fouled, but his free-kick delivery into the area came to nothing.
The Cherries took the lead with their first real attack, with Christie deftly flicking home from Ben Gannon-Doak’s left-wing cross.
David Brooks lifted an up and under that Julio Soler nearly got on the end of - though he was watched all the way by Dom Jefferies.
City threatened when a Tom Bayliss free-kick was headed on by Ladefoged looking for Oyegoke - home keeper Michele Di Gregorio was out quickly to block. The ball landed towards Tyler, but he could not keep his shot down.
The quality of the Premier League side shone through, with Justin Kluivert flicking a header wide and James Hill sending an effort over. Christie also lifted a snapshot wide when the ball rebounded nicely to him in the area.
The Imps threatened down the right with Jefferies pulling the ball back for Hackett, who had the captain’s armband for the night, but a defender got across just time time to nick the ball off his toes.
The last chance of the half went to the Cherries when a corner was cleared only as far as goalscorer Christie, who took a good touch and curled a left-footed shot which Joe Gauci did well to palm away.
The Imps threatened first in the second period, with Jefferies down well to win a throw-in close to the byline despite the best efforts of two defenders. Elder’s long throw was flicked on by Ladefoged but it was easily collected by Di Gregorio.
Bournemouth’s first chance came a few moments later when Alvaro Rodriguez flicked a Gannon-Doak effort just over the bar.
The Cherries had most of the ball, but were limited to shots from distance - with James Hills rifling one over just before the hour mark.
Cohen and Shaw followed with a first double change of the evening as Andrei Counis and Rob Street were sent on.
Street threatened immediately as he almost got on the end of a deep Jefferies cross - the impressive Di Gregorio was out quickly to claim the ball.
The home side also made a change of their own, sending on young Hungary international Alex Toth - with Gauci palming away a free-kick from the 20-year-old almost immediately.
Christie headed home the second goal of the game shortly afterwards, as the Cherries struck from a well-worked corner routine.
Sonny Bradley, Deji Elerewe and Ben House were all sent on in the closing stages as the Imps shuffled their pack.
With the home side also making a number of changes, the game lost some of its flow in the final stages, but Christie did seal his hat-trick by sweeping home the ball after a corner was not fully cleared. Brazil international Rayan, on as a late sub for the hosts, also hit a fourth in time added on.
Ultimately the Cherries deservedly progressed into the fourth round, as a club which continues to show the blueprint for clubs like Lincoln City to continue to compete high up in the football pyramid.
Imps: Gauci, Towler, Hackett, Bayliss, Ladefoged (Street, 58), Jefferies, Elder (Bradley, 73), Honohan (Coubis, 58), Konak (Elerewe, 73), Ogbene, Oyegoke (House, 73). Subs: Wickens, Varfolomeev, Wheatley, Olaofe
Cherries: Di Gregorio, Cook, Hill, Soler, Brooks (Rayan, 78), Christie, Gannon-Doak (Tavernier, 78), Silva (Sanchez, 82), Diakite, Kluivert (Toth, 62), Rodriguez (Jebbison, 78). Subs: Petrovic, Truffert, Scott, Adams.
Referee: Elliot Bell.
