Scotland's defeat at Wembley, in the first competitive meeting between the teams in 17 years, left them six points adrift of Group F leaders England, for which Strachan could pay with his job.
It was Scotland's second successive 3-0 defeat after last month's loss in Slovakia and leaves them facing an uphill battle to qualify for a first major tournament since the 1998 World Cup.
"I feel really proud of the way they played and the way they went about their job," Strachan said.
"If anyone thinks I'm thinking about myself, you're wrong. I've got to think about the guys in there. They've been tremendous for me."
Having preserved their two-point lead at the pool summit, England will approach Tuesday's friendly with Spain hopeful they are turning the page on their Euro 2016 debacle.
The two teams proceeded with a contentious commemorative tribute by sporting black armbands with red poppy motifs.
Britons traditionally wear poppy emblems on November 11 -- Armistice Day -- to honour the country's war dead.
But it leaves England and Scotland open to sanctions from world governing body FIFA, whose rules forbid players from wearing "political, religious or personal slogans, statements or images".
Seeking a change of fortune, Strachan made no fewer than eight changes to his team, which included a first competitive start since 2009 for Celtic goalkeeper Craig Gordon.
England's commitment to playing the ball out from the back gave Scotland hope in the early skirmishes, with Jamie Forrest taking a pot-shot at Joe Hart.
But after Raheem Sterling and Jordan Henderson had seen penalty appeals turned down, England took a 24th-minute lead.
The enterprising Sterling lit the fuse, darting infield before letting fly with a shot that was blocked by Grant Hanley.
Kyle Walker followed up with a fizzing cross from the right and Liverpool striker Sturridge glanced home at the near post to chalk up his eighth international goal.
Scotland procured good chances to equalise either side of half-time. But the unmarked Hanley could only head over the bar from Robert Snodgrass's corner and after Leigh Griffiths cleverly dummied Lee Wallace's cut-back, Forrest dragged wastefully wide.
Scotland were quickly made to lament their profligacy as England doubled their lead in the 50th minute, Lallana leaping to nod Danny Rose's left-wing cross past Gordon.
James Morrison threatened for Scotland, driving into the box and testing Hart with a left-foot shot, but in the 61st minute it was all over courtesy of another headed goal.
Captain Wayne Rooney, back in the starting line-up, swirled a corner towards the near post from the left and Cahill got ahead of his man at the near post to flick a header into the net.
Sterling should have added a fourth, but somehow contrived to hoist the ball over the bar from barely a yard after being picked out by Jordan Henderson's well-struck cross.
The only note of regret for England was a booking for Cahill that will cause him to miss the home game with Lithuania when qualifying resumes in March.
