James Rowe hit the winning penalty after Fon Williams saved spot-kicks from Andre Moritz and Alex Baptiste, after Jermaine Beckford's cool finish had cancelled out Cole Stockton's first-half stoppage-time effort.
Tranmere took a deserved lead when Stockton robbed Zat Knight on the edge of the box and coolly curled past Andy Lonergan.
After half-time Darren Pratley and Tom Eaves forced Fon Williams into a superb 58th-minute double save.
However Bolton's two second-half substitutes combined to see the visitors level as Moritz fed Beckford for a 66th-minute finish.
Evan Horwood cleared Eaves' 84th-minute lob off the line, while Bolton later had an effort disallowed after an offside Eaves tapped in Rob Hall's low cross.
In the first-half of extra-time Fon Williams saved well following Beckford's slick turn and shot, while Eaves had another effort ruled out for offside in the second, before Tranmere's goalkeeper made himself a hero.