Hume went on a splendid solo run in the early stages of extra time before firing in a shot from all of 25-yards from the right hand side which left Bolton keeper Kevin Poole well beaten and despite some late Wanderers pressure the Merseysiders managed to hold on.
Just a minute earlier in normal time debutant Ricky Shakes had diverted the ball past Dutch goalkeeper John Achterberg following a shot by Per Frandsen to save Bolton and to take the game into extra time.
The first half was a very quiet period of the game, Tranmere had failed to test the Wanderers keeper Poole whilst Henrik Pedersen, Mario Jardel and Anthony Barness where reduced to half chances for the home side.
It was only in injury time of the first period when Achterberg was severely tested. Mario Jardel had been brought down by Rovers full back Ryan Taylor and Henrik Pedersen of all people thundered the resulting free kick against Achterberg's right hand post.
The pace of the second half was stepped up considerably and Bolton brought on regular first teamer Kevin Nolan in place of the ineffective Jardel. Nolan's impact seemed to spur on the Premiership side just as it had done in the original tie but Brian Little's team to their credit gave as good as they got.
Nolan almost made an immediate impact following a good lay off from Portuguese youngster Ricardo Vaz Te who was making his first full start in a wide midfield role for the Trotters.
Then with twenty minutes remaining Rovers captain Gary Jones fired just inches over for Tranmere, Poole had to watch the ball dip just over his crossbar and the Merseysiders where being encouraged by their manager Brian Little to push forward and attack Bolton at every available opportunity.
A strong spell of pressure then paid off for the Division 2 side, when on 82 minutes former Aberdeen striker Eugene Dadi took advantage of some indecision in the Bolton defence as he managed to get in between both Charlie Comyn-Platt and Danny Livesey to put Tranmere in front from close range.
However Allardyce then decided to blood young Londoner Shakes for his Bolton debut in the final few minutes and it was a gamble that paid off as in the last minute of normal time Bolton where level as the Academy youngster directed Per Frandsen's shot past veteran keeper Achterberg.
Referee Howard Webb then blew for extra time and the first Tranmere attack proved to produce the winning goal with a strike of some calibre. Hume ran unchallenged by the Wanderers defence and fired an unstoppable shot past the stranded Poole from a good 25-yards out.
It was then Dutch goalkeeper Achterberg who had to be at his best to keep out a shot from Vaz Te and a bending free kick from substitute Nolan as the Trotters stepped up the pressure in a bid to again snatch an equaliser against the Nationwide League opposition.
Then Cleveland Taylor making his first Bolton appearance in two years was booked for infringement as Tranmere held the ball in the corner to waste time. However the young Jamaican was pushed down to the ground and referee Webb had to come between a few fisticuffs, nothing that couldn't be handled and soon the game was back to normal.
As Tranmere tried to run down the clock Bolton tried in vain to get back into the game and their final throw of the dice came when Henrik Pedersen's flying header in the six yard area went wide.
Within a minute the referee had blown for full time and the jinx of Rovers, albeit against a much weaker Wanderers team continues and the Merseysiders will now face Luton Town in the Fourth Round on Saturday week.