The tourists seized the initiative as soon as Elgar won the toss at the Home of Cricket and eventually cruised to an innings-and-12-run victory to go 1-0 up in the three-match series.
The home team side came into the contest on the back of four consecutive Test wins at home but were on the back foot early after being bowled out for just 165 in the first innings.
The Proteas' batters responded with a measured total of 326 all out and that proved more than enough as the bowling unit bundled the hosts out for just 149 in under 40 overs to wrap up a thumping win.
There were a few standout performers for the visitors as paceman Kagiso Rabada underlined his class by getting his name on the famous Lord's honours board with five wickets in the first first innings.
Opener Sarel Erwee struck 73 while Keshav Maharaj and Marco Jansen had vital contributions with both bat and ball but Elgar though was quick to give credit to the entire team.
The South Africa skipper said: "Everyone was brilliant. Sure there's a few guys who had some great individual performances, but I think a lot of guys influenced in moments where there were pressure situations, be it a slip catch or a good stop that influenced the next ball where there was the outcome of a potential wicket.
"I think our batting was pretty solid, especially the middle to late order. They really came out and played the situation to a T.
"That's the kind of awareness that I'm trying to create with the guys and they need to identify those moments and play the game as they see it.
"It's the nature that needs to come and there's lots of learnings still to come for everyone.
"It was a great win. I didn't wake up this morning (Friday) thinking that I'll be doing a presser before five-o-clock. I definitely didn't think that. So yes, it was a pretty special performance and a squad performance."
It was only England's fourth ever innings defeat at Lord's, the Proteas now being responsible for two of those thumping wins including the last one that came by Graeme Smith's lead team in 2003.
Elgar said his team will take the next few days to saviour the victory at the famous venue before switching their attention to the second Test that starts at Old Trafford on Thursday.
He said: "We need to enjoy the moment having achieved this feat at a really unique place, a sold-out Lord's on a Friday, it's pretty big and it's special for a lot of guys who haven't experience that before.
"So we'll enjoy it for the next two days before we focus on the next Test."