LINNETS keeper Jon Higgs made three penalty shoot-out saves on his full first team debut.
Lynn keepers have come into their own in this competition in recent years, Chuck Martini springing to mind, and Higgs became the latest hero following a scoreless extra-time in awful, wet conditions.
Skipper Glen Fuff had earlier cancelled out Banbury's opener.
Higgs, Mark Heath-Preston – also on his full debut – and Adam Jones (back from a hamstring injury to replace the flu-stricken Mark Burrows) were three changes made from Saturday.
Linnets boss Peter Morris said: "Jon had had little to do before that, and went on to do very well.
"Mark has got talent, and I don't think he's 17 yet. I always intended him to play 45 minutes.
"He hasn't been training with the first team – he will be from now on – so he was with players he wasn't familiar, it was very strange for him.
"Considering he was thrown in at the deep end he can be proud of what he did."
Ironically, one of the first things Higgs had to do in normal time was to pick a penalty out of the net in the 23rd minute.
Higgs tried to direct a Paul Raynor back-pass to Jones, but Mark Simms nipped in and Higgs rashly brought him down.
Liam O'Neill put away the spot-kick.
Heath-Preston smashed over and, as The Walks outfit dominated the period soon after the break, Carl Holmes headed wide from Raynor's free-kick.
Lynn levelled on 58 minutes when Fuff struck home from six yards Raynor's chip following a David Staff free-kick.
Holmes and sub Malcolm Ndekwe spurned more chances, while Jack Defty crashed one against the bar with ten minutes left.
In extra-time George Redknap directed an effort over the bar, while at the other end Linnet Mark Camm had a shot rebound off home keeper Simon Tricker.
And so it went to a shoot-out.
Holmes, Richard Woodrow and Staff put away the Linnets' first three penalties, but Redknap and Kieran Sullivan missed to provide Defty with the opportunity to take the tie.
He placed his 12-yard kick straight at Tricker.
However, Higgs' third save put the Linnets through to the second round.
Lynn: Higgs, Jones (Woodrow 90), March, Heath-Preston (Glarvey 45), Fuff, Defty, Camm, Raynor, Bacon (Ndekwe 53), Staff, Holmes. Subs not used: Harvey, Wilson.
Banbury: Tricker, Blossom, Spriggs, Sullivan, Allen, Corbett, Stanbridge, Billington (Stidder 60), Simms (Jenkins 105), Redknap, O'Neill (Breslin 83). Not used: Brock, Taylor.
Referee: Mr Forrester (Luton). Att: 178.