South Africa registered an innings and 12-run victory over England in the first Test.
Such was the dominance of the Proteas team that they crushed the hosts inside three days. The visitors bundled out England for a paltry 165 and 149 across the two innings and posted 326 on the board in the only innings that they batted.
Speaking after the game, Elgar hailed his "special bunch" of players as South Africa strengthened their top position in the World Test Championship (WTC) points table.
"As a playing group it's a special bunch and we play bloody good cricket when we're playing well," Elgar told reporters in the post-match press conference.
Since Dean Elgar’s appointment as red-ball skipper last year, South Africa have won 8 out of their 10 Tests and they are likely to play the WTC final at Lord’s next year.
"I would like to think that what we've laid down as a foundation has been pretty true and pretty solid...it hasn't been fake, it's been real."
Elgar was surprised as his team won the first Test against England in just over six sessions. The second game of the three-Test series will be played at Old Trafford from August 25.
"I didn't wake up this morning thinking I'd be doing a press conference before five o'clock," the 35-year-old said.
While he is aware about the dangers of "complacency", Elgar added: "We need to enjoy these moments.
"We did it at a unique place — a sold-out Lord's on a Friday," the South African captain further said.
Elgar praised pace spearhead Kagiso Rabada, who earned the Player-of-the-Match award for his seven scalps in the game, including a five-wicket haul in the first innings.
"He's an absolute machine when it comes to rising to the occasion for big matches. He put his hand up, he took the bull by the horns and he ran with it."
(With AFP inputs)