Steven Fletcher hit the ground running with a brace as the 10-men Clarets came from behind to kick start their Carling Cup challenge.
Burnley, semi-finalists last season, were in danger of exiting the competition at the first hurdle through Adam Boyd's first half finish.
But second half substitute Fletcher capped a late fightback with the equaliser six minutes from the end of normal time.
Fletcher levels things up
And following Richard Eckersley's controversial sending off in extra time, for a second yellow card, £3m record signing Fletcher headed home the winner to see off the hosts challenge and earn Owen Coyle's men a third round berth.
Coyle had made 10 changes for the Victoria Park tie, with only new signing Andre Bikey surviving the reshuffle.
Among the changes was a first start in over 12 months for Diego Penny in goal, with full debuts for Fernando Guerrero, left back Brian Easton and fellow defenders David Edgar and Richard Eckersley.
And there was a first chance for the Clarets with only 90 seconds on the clock when Guerrero fed Eagles and the former Manchester United ace went on a mazy run through the middle.
Eventually, he squared the ball just behind Steven Thompson, who saw his shot on the turn blocked.
Guerrero headed a bouncing ball over at the far post as the positive start continued.
Hartlepool too were looking to play football and they created their first opening in the 16th minute when Peter Hartley drilled a long ranger into the midriff of a diving Penny.
The Clarets thought they had grabbed the lead midway through the half when Guerrero took advantage of two weak challenges and raced through the middle.
The little Ecuadorian winger could have gone for goal himself, but opted to slide in Rodriguez, who had strayed a yard offside as he stroked the ball home past Scott Flinders.
It was now very much tit for tat, with the Clarets probing and Pools constantly looking to test the reshuffled Burnley back line.
And seven minutes before the break the game burst into life.
Kevin McDonald's cross was inviting for the unmarked Rodriguez, whose attempted back flick at the near post six yards out was totally the wrong option.
And the Clarets instantly paid for that profligacy.
Andy Monkhouse played a neat one two on the left and his little chip into the six yard box was anticipated by Boyd, who raced past a static defence to fire past the exposed Penny.
The Peruvian goalkeeper kept the deficit at one in the 43rd minute, arcing backwards to brilliantly parry Monkhouse's angled drive.
Rodriguez's half chance on the turn right on the whistle was comfortable for Flinders.
And that was the last action of a half that saw the hosts grow in stature as it progressed and rewarded by their interval lead.
The second half began in the same vein, with Pools hustling the Clarets out of possession and playing on the counter.
Denis Behan's sprint down the right was hacked out of the six yard box by Edgar as the hosts threatened a second in the 55th minute.
The hour mark saw Owen Coyle look to the bench and make two substitutions; Fletcher and Martin Paterson replacing fellow striker Rodriguez and Thompson.
And five minutes later Robbie Blake replaced Eagles, instantly occupying a left sided role with Guerrero switching to the right.
And immediately the former Darlington striker was in the action, exchanging passes with Guerrero, but pulling his 20-yard shot wide.
Fellow sub Paterson bobbled a shot on the turn wide with just 16 minutes remaining as another half chance went begging.
And six minutes from time the equaliser finally arrived.
Guerrero's mazy run ended with a half clearance to Eckersley, who fed Blake on the left.
His long diagonal ball was perfection and Fletcher expertly chested the ball outside his marker on the left side of the area and drilled a delicious finish across Flinders.
Suddenly it was all Burnley and McDonald was inches away from converting Guerrero's volley across goal.
And three minutes from time the Ecuadorian could not adjust his feet to find the target when Fletcher's fierce shot three minutes from time was parried by a nervy Flinders.
Four minutes of stoppage time sent shivers through the home ranks, and with 90 seconds left Blake was inches away from winning it with a bit of magic.
The mercurial striker weaved in from the right, played a one-two with Paterson and skimmed the post with a low drive with Flinders beaten.
Extra time was the inevitable result.
And within a minute of the additional 30, the pendulum almost swung back Hartlepool's way.
Colin Larkin found room on the left of the area and crashed a shot off the face of the crossbar.
Penny then saved well from Ritchie Jones down at the foot of his post.
McDonald's shot from the edge of the box was well saved by Flinders as the end-to-end action continued.
And in the 100th minute referee J Moss confused everyone inside Victoria Park by leaving the Clarets to battle out the remainder of the night with just 10 men.
Eckersley got involved in a spot of pull and pushing on the left touchline near the half way line, but nothing that merited a second yellow card, following a first half yellow for blocking of an opponent.
However, Pool's numerical advantage counted for nothing in the 107th minute.
Paterson tricked his way to the by-line and hung up a perfect cross for Fletcher to head home his second from close range.
Back came Pools and as the clock ticked down Jones was inches away from tapping into an empty goal before seeing his goalbound drive cleared off the line by Bikey.
Gudjonsson was then floored twice in as many minutes by fierce free kicks which struck him flush in the face.
But the real pain was felt by Hartlepool as the Clarets held firm to take their place in the next round.
Burnley: Penny, Easton, Bikey, Edgar, Eckersley, Guerrero, McDonald, Gudjonsson (c), Eagles (Blake 65) Rodriguez (Fletcher 61), Thompson (Paterson 61).
Subs: Jensen, Alexander, Jordan, Elliott
Booked: Rodriguez 14, Easton 120
Sent off: Eckersley 100
Hartlepool: Flinders, Haslam, Collins, Liddle, Hartley, McSweeney, Sweeney (Fredriksen 96), Jones, Monkhouse, Behan, Boyd (Larkin 79).
Subs: Cook, Austin, Rowell, Greulich, Power
Booked: Sweeney 23, Collins 45
Ref: J Moss
Att: 3,501 (621 Burnley)