Substitute Jorgen Strand Larsen scored twice in two minutes as Wolverhampton Wanderers beat fellow Premier League strugglers West Ham United 3-2 to reach the third round of the League Cup on Tuesday.
Local hero Burn gave Newcastle a deserved lead in the 45th minute when the defender thumped in a header from a corner.
The two teams had numerous chances in the first 90 minutes.
Guardiola said he would not take a risk with his leading goalscorer when they host Sevilla in Europe's top-tier club competition as they have already qualified for the knockout stage.
Erik ten Hag's side, featuring several players who played in the World Cup in Qatar, could have doubled their lead when Anthony Martial's shot was well saved by Burnley goalkeeper Bailey Peacock-Farrell.
Holders Manchester City were eliminated from the League Cup by West Ham United after a penalty shootout at London Stadium on Wednesday, ending Pep Guardiola's team's four-year domination of the competition.
Koopmeiners leads Atalanta past Milan into Coppa semis
Marcus Rashford continued his fine season as he scored again to help Manchester United beat Nottingham Forest 3-0 in their League Cup semi-final first leg on Wednesday.
Manchester United will be back in action in next season's Champions League after they hammered Chelsea 4-1.
Manchester City beat League Cup holders Liverpool 3-2 in a cracking game on Thursday, taking the lead three times before holding on to reach the quarter-finals.
Manchester United crushed Newcastle United's hopes of a claiming a first domestic trophy for nearly 70 years with a clinical 2-0 victory in the League Cup final at Wembley on Sunday.
Rashford also credited former team manager Jose Mourinho for making him a better player.
Guardiola finally has chance to do justice on European stage: Sandhu
Arsenal squeezed past Liverpool 5-4 on penalties to advance to the League Cup quarter-finals on Thursday after a goalless 90 minutes.
Chelsea have completed the transfer of defender Wesley Fofana from Leicester City, both Premier League sides said on Wednesday, with the Frenchman signing a seven-year contract with the west London club.
Manchester City won the English League Cup for the second successive season as the beat Chelsea 4-3 in a penalty shootout after the Wembley showpiece had ended 0-0 after extra time.
Barring a monumental collapse Manchester City will win at least two trophies this season, but former England captain Alan Shearer believes European bragging rights are what the club's Abu Dhabi owners crave most.
Liverpool beat Arsenal at Anfield in a penalty shootout on Wednesday after their last-16 tie ended 5-5 in regulation.
Liverpool held their nerve to win an astonishing penalty shoot-out 11-10 and edge out Chelsea in the English League Cup final on Sunday after a rip-roaring contest had somehow ended 0-0 after extra time at Wembley.
Timo Werner scored his first goal of the season as a much-changed Chelsea side crept into the fourth round of the League Cup by beating Aston Villa 4-3 on penalties after the Stamford Bridge clash ended 1-1 on Wednesday.
Manchester City reached their fourth straight League Cup final as second-half goals from John Stones and Fernandinho gave Pep Guardiola's side a 2-0 win over Manchester United in Wednesday's semi-final at Old Trafford.
'It's not easy to handle a team at a big, big club... always demanding to win competitions and every manager needs time. I have the feeling United started to play the way he wants'
Chelsea manager Maurizio Sarri is uncertain if Kepa Arrizabalaga will start Wednesday's English Premier League match against Tottenham Hotspur after the goalkeeper's refusal to be substituted in their League Cup final defeat by Manchester City.
Manchester City manager Pep Guardiola defiantly wore a yellow ribbon on his suit in support of jailed pro-independence Catalan politicians during his side's 3-0 League Cup final victory over Arsenal at Wembley Stadium on Sunday.
Theo Walcott should leave Arsenal for his own good after being relegated
Pep Guardiola has criticised the weight of the match ball used in the League Cup after his team failed to score in their fourth round match against Wolverhampton Wanderers and won 4-1 on penalties.
Manchester City continued their imperious stranglehold over the League Cup with a 2-1 victory over Aston Villa with goals from Sergio Aguero and Rodri in the final at Wembley on Sunday to win the trophy for a third straight season. City became the first side to win the competition three seasons in a row since Liverpool claimed four straight titles from 1981 to 1984.
Arrizabalaga appeared to be injured as he prevented City's Sergio Aguero from scoring in the closing moments of extra time, and manager Maurizio Sarri ordered his substitution before Chelsea went on to lose the showpiece match 4-3 on penalties.
Aymeric Laporte's late header sealed a deserved 1-0 victory over Tottenham Hotspur
Reaction to Arrizabalaga's refusal to be substituted
United looked more solid after the introduction of midfielder Nemanja Matic at the break and got a foothold in the tie through a 70th-minute goal from Marcus Rashford.
Manchester United's current malaise was laid bare as they struggled past third-tier Rochdale 5-3 on penalties after a 1-1 draw in the League Cup third round.
Sergio Aguero and Kevin De Bruyne put on an attacking masterclass to help City rout Leicester City 5-1.
A summary of Saturday's matches in the FA Cup.
News of all that's transpired on and off the football field
The father of four and founder of duty-free King Power International was a huge favourite with the club's fans.
Pep Guardiola claimed his maiden English silverware as his side thrashed Arsenal 3-0 in the League Cup final with Sergio Aguero, Vincent Kompany and David Silva all on target in a disappointingly one-sided showpiece on Sunday.