In-form Charlie Austin seemed to have fired Burnley into the third round with a first-half winner.
The former Swindon striker, who rattled in 17 goals last season, took his tally to four in five this term with a well-taken header eight minutes before half-time.
But Plymouth, who claimed their first win in the competition in five years when knocking out Portsmouth in the first round, stunned the Clarets by equalising in injury time.
Sam Vokes was penalised for a foul on Paris Cowan-Hall and Robbie Williams tucked home from the spot to force extra-time.
Vokes had a couple of opportunities to redeem himself but failed to take them as Plymouth continued to frustrate their opponents.
Conor Hourihane then had a late sight of goal as Plymouth hunted a shock at the end of extra-time but Jensen was equal to the task.
The long-serving Dane then took the plaudits in the shoot-out as he saved from Williams, Luke Young and Hourihane, leaving Brian Stock to convert the decisive spot-kick as the Clarets prevailed 3-2.