The hosts scored a controversial late goal to half the deficit but could not find an equaliser to force a replay.
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Aden Flint had an early effort cleared off the line as Bristol started stronger than their Conference opponents.
The Robins took the lead 10 minutes before half time when top scorer Emmanuel-Thomas let fly from distance and found the top corner.
They almost doubled their lead soon after the break as first Sam Baldock was denied by a last-ditch Tony Capaldi tackle and then Scott Wagstaff missed an open goal from distance after Baldock had closed down the Tamworth keeper.
Baldock got his goal with seven minutes left on the clock by finishing with a clipped finish after Ben Richards-Everton failed to deal with a long ball.
Andy Todd was awarded a late goal after scrambling in a long throw-in but it appeared to not quite cross the line.
The Robins will now become ball 63 in the third-round draw, which is due to take place shortly.
You can follow live text coverage here on Sports Mole as the teams from the Premier League and the Championship enter the competition.