Sebastien Haller eased West Ham's woes with a double in their 3-0 victory over Charlton in the League Cup second round, while Crystal Palace were knocked out by Bournemouth after a marathon penalty shoot-out on Tuesday.
French striker Haller scored twice in four minutes midway through the first half and Felipe Anderson added a late third for David Moyes' troubled side.
It was a much-needed result for the Hammers, who face Leeds or Hull in the third round, after they started the Premier League season with a dismal 2-0 home defeat against Newcastle on Saturday.
That limp surrender came after fan protests against co-owners David Gold and David Sullivan, while West Ham captain Mark Noble criticised the club's lack of ambition following the sale of promising midfielder Grady Diangana to West Bromwich Albion.
With Arsenal, Wolves, Leicester, Tottenham, Manchester City and Liverpool on West Ham's immediate fixture schedule, it was vital to avoid an embarrassing exit against third-tier Charlton at the London Stadium.
"Probably the only disappointment is we should have won more comfortably," Moyes said.
"But Seb getting a couple of goals will be good for his confidence. We had Michail Antonio scoring for us at the end of last season and now we hope Seb gets on a run of scoring goals."
Jack Wilshere was absent from the West Ham squad despite 10 changes made by Moyes, with the club reportedly wanting to sell the injury-prone midfielder.
Bournemouth goalkeeper Asmir Begovic missed a spot-kick but still finished as the hero in the Championship club's 11-10 penalty shoot-out win over Premier League side Crystal Palace.
After a goalless 90 minutes at the Vitality Stadium, Begovic saw his penalty saved by Wayne Hennessey.
It was the first miss out of 21 penalties but, after Hennessey blazed over the crossbar, Begovic saved from Palace captain Luka Milivojevic to give Bournemouth a trip to holders Manchester City.
- Grealish stars -
Jack Grealish celebrated his new Aston Villa contract by scoring in their 3-1 win at League One side Burton Albion.
Villa's club-record signing Ollie Watkins equalised for the Premier League team late in the first half to cancel out Colin Daniel's opener after only 68 seconds.
Grealish, who signed the new five-year deal earlier on Tuesday, struck a sweet volley from the edge of the area in the 88th minute.
Keinan Davis's stoppage-time goal ensured last season's League Cup runners-up head to Bristol City or Northampton.
Ryan Fraser marked his Newcastle debut with the winner as the Premier League side beat Championship club Blackburn 1-0 at St James' Park.
Magpies boss Steve Bruce made 10 changes from the win at West Ham and former Bournemouth winger Fraser took his chance with the 35th minute goal that earned a trip to fourth tier Morecambe.
Tottenham striker Harry Kane will face the team whose shirts he sponsors after Leyton Orient beat Plymouth 3-2 to earn a clash with Jose Mourinho's men.
Championship side Barnsley will travel to Chelsea after first half goals from Patrick Schmidt and Jordan Williams sealed a 2-0 win at Middlesbrough.
Luton booked a home tie against Manchester United after beating fellow Championship team Reading 1-0 thanks to Jordan Clark's 24th minute header.
Third tier Lincoln will host Premier League champions Liverpool after thrashing Bradford 5-0.
It will be the first meeting between the clubs since 1961.
All the third round ties are due to be played next week, with the dates to be set later.