McLeish, who guided Birmingham to the trophy last season, could have few complaints at the outcome at Villa Park as Eagles and Kakuta confirmed the Trotters' superiority.
Bolton boss Owen Coyle made nine changes for the third-round clash but his side were still dominant in inflicting a first defeat of the campaign on a Villa side which was close to full strength.
The final whistle was the signal for loud boos from the Villa fans in the 22,261 crowd.
Only Villa striker Gabriel Agbonlahor posed a threat for the visitors whose goalkeeper Adam Bogdan was not seriously tested until the 87th minute.
Skipper Stiliyan Petrov lacked support in midfield and Bolton looked hungrier and more composed as they sealed a place in the last 16.
Bolton made a positive start and the first chance fell to Kakuta after Eagles had outpaced Alan Hutton and sent a low ball across the box.
But Stephen Warnock was perfectly placed to block the close-range effort from the on-loan Chelsea player.
Shay Given then went down to his right to cling onto David Ngog's long-range drive.
The home side's first meaningful attack saw the in-form Agbonlahor tee-up Nathan Delfouneso on the edge of the area.
But Bolton defender Gretar Steinsson was alert to the situation and tidied up.
The home side tried to gain a foothold in the game and a Barry Bannan cross picked out Agbonlahor's run into the box. But the England striker failed to make a clean contact on his header.
Bolton still looked composed going forward and the woodwork came to Villa's rescue after 35 minutes.
Eagles' corner was met by Trotters centre-back David Wheater and his header thudded against the outside of a post.
Stuart Holden made a crucial near-post stop to block Agbonlahor's shot from Albrighton's dipping centre.
Former Villa defender Gary Cahill became the first player to be yellow carded five minutes before half-time for bringing down Stephen Ireland.
Villa enjoyed a spell of pressure and Cahill blocked James Collins' header from a Bannan centre.
Delfouneso should have done better with his header from another Bannan centre but failed to hit the target.
Agbonlahor remained Villa's most potent threat but after 54 minutes Eagles gave Bolton the lead.
The former Manchester United youth player took advantage of the ball bouncing off Albrighton into his path and hammered a powerful drive past Given from 20 yards out.
Eagles found himself in a similar position two minutes later but this time his shot was deflected past the post as Villa looked at sixes and sevens.
McLeish made his first change after 71 minutes with Charles N'Zogbia replacing Albrighton while Coyle brought on Kevin Davies for Ngog at the same juncture.
But, with 12 minutes, left Kakuta doubled Bolton's lead.
Ricardo Gardner floated a free-kick into the box and Kakuta had time to drill a powerful shot past Given.
Bannan was booked for a challenge on Gardner with four minutes remaining.
Bogdan made his first meaningful save after 87 minutes when a shot from Collins bounced off his chest.