Tamim celebrated his 23rd birthday with a 57-ball 59 and Shakib hit a 46-ball 56 as the hosts achieved the rain revised target of 212 with 17 balls to spare in the day-night match, watched by Bangladesh Prime Minister Sheikh Hasina.
Bangladesh's victory -- only their third over Sri Lanka in 30 one-day internationals -- sees them through to meet Pakistan in the final on Thursday.
Pakistan topped the league with nine points from three matches, followed by Bangladesh (8/3), India (8/3) and Sri Lanka (0/3). The hosts made it to the final as they beat India in their league match.
"The final is one of the biggest achievements in our cricketing history. Hopefully, we will cross that hurdle. It is a great boost to have the crowd supporting you," said Bangladesh captain Mushfiqur Rahim.
"It has been a great tournament for us. Nazmul Hossain did well and that set up the game. We had a stiff target after the rain, but Shakib and Tamim batted well. There are five or six people performing in every game."
Bangladesh were struggling at 40-3 following two wickets from paceman Nuwan Kulasekara before Tamim and Shakib put on 76 for the fourth wicket to raise their team's hopes.
Tamim was caught by Lahiru Thirimanne at point while attempting to cut off-spinner Sachithra Senanayake after hitting nine fours in his third successive half-century.
The hosts suffered another setback when Senanayake dismissed well-set Shakib, trapped leg-before after hitting seven fours.
But Nasir Hossain (36 not out) and Mohammad Mahmudullah (32 not out) added 77 runs for the unfinished sixth-wicket stand to steer their team home, much to the delight of a capacity crowd at the Sher-e-Bangla stadium.
Bangladesh earlier put in a disciplined bowling performance to restrict Sri Lanka to a modest total, with paceman Nazmul (3-32) and left-arm spinners Abdur Razzak (2-44) and Shakib (2-56) being the main wicket-takers.
"We did not play to our potential," said Sri Lanka skipper Mahela Jayawardene.
"Bangladesh played very well today and they created opportunities from the start. It was a team effort from Bangladesh. It's always going to be tough when you play Bangladesh at home."
Nazmul rocked Sri Lanka in his sharp six-over opening spell, removing Jayawardene (five), Tillakaratne Dilshan (19) and Kumar Sangakkara (six) to send the tourists reeling at 32-3.
Chamara Kapugedera (62) and Thirimanne (48) steadied the innings with an 88-run stand for the fourth wicket, but were not allowed to score freely by the Bangladeshi bowlers.
Upul Tharanga was the other main scorer, scoring a 44-ball 48 with one six and five fours.
Fast bowler Mashrafe Mortaza was the other bowler to impress, conceding only 30 runs in his tight 9.5 overs.