A first-half strike from on-loan Tottenham midfielder Massimo Luongo looked to have won it for Ipswich until Beck came up with a 90th-minute header to send the second-round tie into extra-time.
Symington then grabbed the winner in the first period of extra-time.
Ipswich showed no ill effects of their six-goal hammering at Blackpool on Saturday as they set the Carlisle defence a busy challenge in the opening 45 minutes.
Twice keeper Adam Collin kept Carlisle in the game with last-ditch saves, the first time using his legs to thwart Josh Carson from close range then getting the better of Michael Chopra one on one when the striker eluded Peter Murphy to race clear.
Ipswich took the lead midway through the first half with a raking shot from just outside the penalty area by on-loan Spurs midfielder Luongo.
This time Collin had no chance of getting near to the ball as it fizzed past him into the net.
The home side more than held their own against their fading Championship opponents in the second half as Beck and Paddy Madden went agonisingly close in a series of late attacks.
It was 18-year-old sub Beck who sent the game into extra-time with a flashing header in the 90th minute after Matty Robson had crossed from the left.
Symington the lashed the ball home from just inside the area after latching on to Beck's pass to give his side a deserved 99th-minute lead which they held to take an upset victory.