Emmanuel-Thomas made it 15 goals for the season with a 25-yard thunderbolt before the break and Baldock's cool finish sealed victory, although Andy Todd pulled one back for the Lambs late on.
Tamworth survived an early scare as Tony Capaldi scrambled a fourth-minute Aden Flint header off the goal-line, but Emmanuel-Thomas' rocket broke their resistance nine minutes before the break.
The Sky Bet League One strugglers should have secured progress early in the second half, but Baldock fired two good openings wide and saw a goalbound shot denied by Capaldi's last-ditch block.
Baldock's industry created another chance as he blocked a clearance from Lambs keeper Cameron Belford but, with an open goal gaping, City forward Ryan Taylor steered wide from 30 yards.
City should have been made to pay for those misses soon after as Tamworth striker Justin Richards raced clear, but he fired wide with only Robins keeper Elliot Parish to beat.
Baldock seemed to have put the tie to bed in the 83rd minute, sprinting through to drill beyond Belford, and although Todd bundled home to give Tamworth a lifeline in the 90th minute, City held on.