Richard Kalwarczyk went close in the early stages, forcing a fine save from Tom Monger and then seeing his effort on the rebound cleared off the line by Ashley Saunders.
Tadley’s best chances of the first half fell to top-scorer Kye Theodore, who twice forced New Inn goalkeeper Dean Mahoney into decent blocks.
Theodore had an even better chance on the hour. The striker was put through by Stephen Burgess but Mahoney made a fine save to deny him.
Matt Bennett almost won the game for New Inn in normal time, but the substitute’s effort crashed against the crossbar, while Simon Penfold headed inches wide for Tadley and the sides were forced to play an additional 30 minutes.
The game seemed to be heading towards a penalty shoot-out until 10 minutes before the end, when Kalwarzck fired into the top corner from 25 yards.
Tadley looked to reply, with Theodore firing inches over and Craig Farrington’s free kick being blocked on the line, but New Inn made the game safe at the death, Luke Richardson heading home Bennett’s corner.