The pair will meet in a replay on Tuesday December 3 to decide who faces Sheffield United in the second round after neither could break the deadlock.
Bury's Danny Mayor blasted wastefully over the top, while Appiah headed wide from a Liam Hughes free-kick.
The second half began in a much more energetic fashion, Harrison Dunk nodding wide from Richard Tait's cross.
Jensen then made a superb reaction save to turn away Appiah's shot from a Hughes cross as Cambridge piled on the pressure.
Bury substitute Tommy Miller had a powerful shot cleared off the line by Tait but Cambridge had the better chances in the closing stages.
Jensen made another excellent save to tip a curling effort from Dunk over the bar.
- Kwesi Appiah
- Liam Hughes
- Sheffield United
- Richard Tait