City were held to a 1-1 draw at League One Rotherham, before securing a bonus point with a 4-2 win on penalties.
Spurs' under-21s earned a point in a 2-2 draw with the Hatters, who beat the visitors 4-2 in the shootout.
League One side Plymouth also won on penalties, beating Chelsea's development squad 5-4 after a 2-2 draw.
There were wins for two Premier League development squads inside 90 minutes, as West Ham and Swansea both earned three points in the opening round of fixtures.
Swindon twice equalised against the Hammers after falling behind, but Dan Kemp slid in a third goal late on to secure the visitors victory at the County Ground.
Kyle Storer gave League Two side Cheltenham the lead early in the second half against the Swans, but Kenji Gorre equalised in the 73rd minute, before Aaron Lewis slotted in an 83rd-minute winner.
Notts County were the only English Football League side to earn the maximum three points, as goals from Jonathan Forte and Haydn Hollis earned the Magpies a 2-1 win over Everton Under-21s.
Swindon Town 2-3 West Ham United Under-21s
Plymouth Argyle 2-2 Chelsea Under-21s (Plymouth won 5-4 on penalties)
Cheltenham Town 1-2 Swansea City Under-21s
Luton Town 2-2 Tottenham Under-21s (Luton won 4-2 on penalties)
Rotherham United 1-1 Manchester City Under-21s (Manchester City Under-21s won 4-2 on penalties)
Notts County 2-1 Everton Under-21s
Manchester City Under-21s won on penalties in their first EFL Trophy appearance, while fellow debutants Tottenham lost in a shootout at Luton.