The striker headed home Richard Taundry's throw in the 85th minute after Jamie Taylor's impudent first-half chip looked set to send the Imps through to the second round.
Dean Holden whistled an effort just over for Walsall after 12 minutes before Lincoln began to threaten and Power thumped the bar from 15 yards and Mills' follow-up was turned wide by David Grof.
Taylor capped an otherwise forgettable first half when he delightfully chipped over Grof from an angle in stoppage time to fire the Imps ahead.
The brave Blue Square Premier hosts deserved their lead and Grof's smart stop denied Taylor a second 11 minutes after the break.
Alan Power struck the bar for the hosts while Walsall offered little before Bowerman's seventh goal of the season spared the League One side's blushes.