Daniel Sturridge (left) scored England's opening goal against MaltaDaniel Sturridge (left) scored England's opening goal against Malta (AFP Photo/Ian Kington)
Southgate has stepped into the breach following the departure of Sam Allardyce, who was brought down by a newspaper sting, and saw his side secure a successive World Cup qualifying victory.
Sturridge and Alli struck within nine minutes of each other at a nearly full but subdued Wembley, but England were unable to swell their lead despite monopolising possession.
"I thought first half we played well. We could and should have scored more goals," said Southgate, whose side were booed by some fans at the final whistle.
"Their goalkeeper made some good saves and subconsciously I think some of our players looked after themselves physically in the second half.
"We missed a couple of good chances, but given where we were 10 days ago, I am pleased with the result."
Arsenal left-back Kieran Gibbs was consequently called into the squad ahead of Tuesday's next Group F match in Slovenia.
Southgate will also preside over November's qualifier against Scotland and friendly with Spain before the Football Association decides who will succeed Allardyce on a full-time basis.
Malta's goalkeeper, Andrew Hogg, hails from London and it was no surprise he was a central figure on his return to the city of his birth.
He was called into action for the first time in the eighth minute, plunging to his right to palm away a curling effort from Wayne Rooney.
Rooney, deployed in central midfield, was upstaged by his midfield partner Jordan Henderson, whose enterprising running and incisive passing kept Malta on the back foot in the first half.
His lofted pass freed Lingard, whose mishit half-volley flashed over, and he teed up Alli for a header that was parried by Hogg, with Sturridge wastefully blazing the rebound over.
- Made in Liverpool -
It was not all plain sailing for England.
Bertrand was forced off with an apparent hamstring injury in the 19th minute, which saw Danny Rose come on, while Rooney was lucky to avoid sanction for an untidy challenge on Malta skipper Andre Schembri.
England took the lead in the 29th minute with a goal made in Liverpool, Henderson swishing a cross into the box from deep and Sturridge steering a header inside the right-hand post.
It was Sturridge's 100th goal in professional football.
After Hogg denied Lingard and Gary Cahill sliced wide from a corner, England inflated their lead in the 38th minute.
Southgate's men circulated the ball patiently before Henderson lit the detonator, charging to the edge of the box and then ceding possession to Alli.
The Tottenham Hotspur man's initial attempt was repelled by Hogg, but he followed in to prod home his second international goal.
Hogg saved from Sturridge, twice, and tipped a Rooney free-kick over the bar early in the second half before sliding out to block from Theo Walcott, who had been slipped in by Lingard.
Walcott and Sturridge gave way to Marcus Rashford and Jamie Vardy, the Manchester United youngster and Leicester City striker both getting rousing receptions from the home fans.
Shy of a Lingard shot that flew wide, England struggled to create chances as the game wore on.
Malta's Andrei Agius almost made things interesting with a late volley that England goalkeeper Joe Hart had to turn behind.
Alli should have made the game safe when he was picked out by Rashford, but from barely six yards out he sent the ball over the bar.
"England are stronger than us, but we stayed in a good shape all the game," said beaten manager Pietro Ghedin.
"We saved a lot of chances. We suffered a lot, but we played a good game."