It took half an hour for the former England defender's selection to break down a stubborn rearguard but thanks to first-half goals from Daniel Sturridge and Dele Alli they did go some way to meeting Southgate's demands.
However, in the first outing since Sam Allardyce's embarrassing departure as manager after one match and 67 days in charge, the victory that gave England maximum points at the top of the group brought only the odd smile to a Wembley full house of 81,781.
Southgate, who has four games to secure the job full-time, had a right to feel irritated that, despite completely outclassing a side ranked 176th in the world, his team did not have the wit to make their second-half domination count with a hatful of goals.
Starting with captain Wayne Rooney in midfield and giving a debut to Jesse Lingard, a former favourite when he was in charge of England's under-21 team, Southgate was hoping for sharper finishing.
In the first half there was a notable shift in tempo and intensity from the side that performed sluggishly in their one match under Allardyce.
Malta packed the defence and England initially lacked ideas, Rooney going closest with a curling shot well-saved at full stretch by London-born keeper Andrew Hogg.
SURGING RUN
Jordan Henderson then crossed deep after 29 minutes and Sturridge's well-placed header found the corner of the net.
England made it 2-0 when Henderson's surging run in the 38th minute resulted in the ball falling to Alli who scored at the second attempt.
Malta remained content to let England huff and puff after the break and, apart from one free kick from Rooney and a close-range attempt by Theo Walcott that were well saved, the fans had little to cheer.
The most rousing reception came when Marcus Rashford, unlucky not to get a start, was brought on for the last 23 minutes.
The young Manchester United striker made an impact with a neat ball to Alli but his effort ballooned over the bar.
It took Malta 80 minutes to offer a goal threat and Rooney, after an ordinary performance, received a few boos in the dying seconds after looping a cross high and long.
England play their third qualifier in Slovenia on Tuesday.
(Editing by Tony Jimenez)
