The 22-year-old left-hander smashed an unbeaten 127 as the fired-up hosts surpassed Pakistan's modest 250 all out in the 40th over at the Sher-e-Bangla stadium.
Pakistan captain Azhar Ali's 100 made no difference to his team's fortunes after Bangladesh had won the first match by 79 runs and the second by seven wickets.
It was the first series whitewash by the Tigers against an Asian Test-playing nation, and followed their appearance in the quarter-finals of the recent World Cup in Australia and New Zealand.
Bangladesh captain Mashrafe Mortaza said he was delighted at the way the team had shaped up in the past few months and hoped to continue the good work in future.
"This is a good team we have and I am really happy the way we are playing," he said. "Soumya (Sarkar) was brilliant today but our work is not done yet. We will now focus on the remaining matches."
The two teams will play a Twenty20 international on Friday before starting a two-Test series from April 28.
- Dramatic collapse -
=====================
The win was never in doubt after Sarkar, who hit 13 boundaries and six sixes in his 110-ball innings, put on 145 for the first wicket with Tamim Iqbal.
Tamim, who had slammed centuries in the first two matches, missed a third consecutive hundred when he was leg-before to Junaid Khan for 64.
The left-arm seamer also bowled Mohammad Mahmudullah in his next over, but Mushfiqur Rahim (49 not out) ensured victory with an unbroken stand of 97 for the third wicket with Sarkar.
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