The former Under-21s head coach is seeking a third successive victory at the helm this week, having masterminded 2-0 wins over the Republic of Ireland and Finland during September's international break.
Declan Rice and Jack Grealish were on target in Carsley's first game in charge, before a double from Harry Kane on his 100th England appearance saw the Three Lions make it two for two against Finland.
Should England manage to get the job done against Greece on Thursday, Carsley would become just the second-ever England men's manager to win their first three competitive matches in charge after Fabio Capello.
The Italian did lose his second game in charge in 2008 - a 1-0 defeat to France in a friendly - but each of his first eight competitive games as England manager ended in victory, all of which came during 2010 World Cup qualifying.
Carsley chasing England history in Greece clash
However, England only kept one clean sheet in their first three competitive wins under Carsley, shutting out Andorra in a 2-0 win before a 4-1 beating of Croatia and 5-1 thumping of Kazakhstan.
As a result, Carsley could now become the first-ever Three Lions boss - temporary or permanent - to win each of their first three competitive matches in charge without conceding a single goal.
The 50-year-old has already been handed a significant boost in his quest for that new record, as Greece are missing one of their key attackers due to injury, Panathinaikos star Fotis Ioannidis.
The 24-year-old scored three of Greece's five goals last month, a brace against Finland before a further strike against Ireland, and only Slovenia's Benjamin Sesko (four) boasts more in League B so far.
England are not without their own offensive concerns, though, as Carsley is sweating over the fitness of star striker Kane, who is still training individually after suffering an injury while playing for Bayern Munich last weekend.
Kane could join exclusive England club against Greece
The Three Lions' all-time leading scorer has not been ruled out of Thursday's game just yet, though, and he will have the chance to join an exclusive club of England goalscorers should he play.
Kane has found the back of the net in each of England's last six competitive home games, and he is now bidding to become just the third Three Lions player to do so seven times in a row.
Only two Englishmen have ever scored in seven straight competitive home games, most recently Wayne Rooney from 2012 to 2015, while Steve Bloomer - Derby County's all-time record goalscorer - did so from 1895 to 1899.
England and Greece are locked together on six points at the top of League B Group 2, although the Pirate Ship are leading the way thanks to a marginally superior goal difference.
Thursday's game will be the first showdown between England and Greece since a 2006 friendly, which the Three Lions won 4-0, and the hosts have never lost to the 2004 European champions in nine previous meetings.