Bruno Berner, Matty Fryatt and James Wesolowski were successful from the spot for the Foxes with Andy King missing the target after the sides played out a goalless 90 minutes.
For League Two Lincoln, only Lee Frecklington was successful with his spot-kick as Scott Kerr and Danny N'Guesson were both denied by Henderson before former Leicester defender Frank Sinclair blazed his penalty over the bar.
During normal time, Ayden Duffy's sprawling save denied Fryatt after 14 minutes before the home side almost broke the deadlock on the half-hour mark when Nicky Adams freed Fryatt down the right and Steve Howard fired the resulting cross just wide.
Lincoln's best chance came just before the break when N'Guesson's 20-yard angled drive appeared destined for the top corner of the net until Henderson produced a brilliant one-handed save.
Late on, Fryatt squandered a fine opportunity to settle the tie but, after exchanging passes with Adams, he shot straight at Duffy from eight yards.