Chasing 271 runs to win, Victoria were bowled out for 266 with Bell's 6-72 including the last two wickets on a dramatic final day at Bellerive Oval.
The Tasmanians were in the box seat when Victoria slipped to 7-171.
But wicketkeeper-batsman Liam Blackford, in his second first-class game, made a classy 80 and combined with Fergus O'Neill (53) to take the Victorians to the cusp of victory.
Their 81-run partnership for the eighth wicket ended when the 22-year-old Blackford was dismissed with 19 required.
Tasmania's Bell took the second new ball and O'Neill was soon aghast to be caught on the deep point boundary with just eight more runs needed.
Victoria's second-gamer Harry Hoekstra and No.11 David Moody could muster just three more runs before Bell struck the final blow when Hoekstra edged to 'keeper Jake Doran.
Tasmania captain Jordan Silk described the finish as chaotic.
"You always want to be on the right side of those," Silk said.
"It's funny, we have probably played a few of them in the last few years and taken a little bit of belief from winning those close ones in the past.
"(I) just always had that weird belief that we would get things done, even when it probably actually swung towards their favour.
"(Coach) Jeff (Vaughan) referenced showing a bit of backbone - we showed plenty in this game to dig ourselves out of multiple holes."
Victoria resumed at 5-98 and veteran Marcus Harris and Blackford were largely untroubled in moving the score to 149 when their 53-run union was broken.
Harris (63) attempted a late cut from the bowling of Marcus Bean (2-62) but edged to Silk at wide first slip.
The dismissal left the Victorians still requiring 122 runs for victory but Blackford reduced the target by seven with just one shot.
The left-hander cut to the near deep point boundary and ran three with Will Sutherland. On taking the return throw, 'keeper Doran aimed at the bowler's end stumps but missed, with the ball travelling for four overthrows.
Tasmania's hero Bell soon removed Sutherland for three with a superb delivery that beat the batter's outside edge and clipped the top of off stump.
Blackford and O'Neill then formed their partnership but ultimately fell short as the Victorians slipped to consecutive defeats to open their Shield campaign, with Tasmania logging their first win of the season.
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