England, at lunch on the first day of this four-match series, were 71 without loss.
Johannesburg-born Strauss, in his 50th Test, was 26 not out and fellow left-hander Cook 39 not out, with South Africa unable to take the early wickets captain Graeme Smith would have wanted after electing to field first having won the toss.
South Africa quicks Dale Steyn and Makhaya Ntini kept things tight early on with England managing just three runs in the first seven overs.
But Smith, curiously, withdrew Steyn from the attack after an impressive burst of three overs for one run with two maidens.
Strauss, on his Middlesex home ground, broke the shackles in the eighth over when he pulled Ntini for the first four of the match.
Cook then took advantage of width from both first change Morne Morkel and Ntini with a couple of well-struck square-cut boundaries.
Smith shuffled his pack and the introduction of all-rounder Jacques Kallis meant four pace bowlers had been used in the initial 11 overs.
The 6ft 6in Morkel, coming round the wicket, struggled with his line in his first Test on the eight-and-a-half foot slope which runs across Lord's and that negated in part the 92mph pace he was generating while a cross-wind did little for his accuracy either.
Although the acrobatic Mark Boucher prevented a couple of deliveries going for bonus runs, he could do nothing about one especially wayward ball which, somewhat harshly for the wicket-keeper, was signalled four byes and not wides by Australian umpire Daryl Harper.
Strauss seized on a rare loose ball from Kallis, cutting it off the back foot for four.
Morkel though did start to adjust to the conditions and beat the outside edge on several occasions and England's fifty came up when Cook edged Kallis through the slip cordon for four.
With South Africa's pace quartet unable to make a breakthrough, left-arm spinner Paul Harris was brought on shortly before lunch which Cook had little problem in playing out for a maiden.