England, at the fourth day's close, were 258 for seven in their second innings after Strauss had scored 128.
Pakistan's Test record fourth innings total to win was the 315 for nine they achieved against Australia at Karachi in 1994/95.
And the equivalent Lord's record was West Indies' 344 for one, requiring 342, set 22 years ago.
Leg-spinners Danish Kaneria (three for 73) and Shahid Afridi (one for 59) kept hopes of a successful chase alive with tight bowling on a wearing pitch.
Left-handed opener Strauss, leading the side because of Andrew Flintoff's ankle injury, became only the third England player, after Archie MacLaren in 1897/98 and Allan Lamb in 1989/90, and first at home, to score a hundred in his debut Test as England captain.
His ninth Test century was also his third at his Middlesex home ground after he's scored 112 here on debut against New Zealand at Lord's in May 2004 and 137 against the West Indies two months later.
"It was a tricky situation as we were losing wickets at important times and we couldn't push on," said Strauss.
"We needed someone to stick around and fortunately I was able to do that.
"I've had a frustrating time this year, getting 20s and 30s isn't good enough for Test cricket. It was a case of overcoming that, the wicket here was flat and it was nice to get to three figures."
Strauss said that he will sleep on a decision on whether or not to declare first thing on Monday, the final day of the match.
"The wicket is flat so we have to take that into consideration. We'll have to bowl well. The ball bounced a little for their spinners today. Hopefully (off-spinner) Monty Panesar can get that and be a threat.
"The new ball is when the variable bounce is at its greatest. If it's going up and down Steve Harmison is going to be hard work."
Meanwhile Pakistan coach Bob Woolmer said England were favourites to win.
"I don't think any of our team play to survive," the former England batsman told reporters.
"They'll play the way they play best and if England bowl badly and we get off to a flyer, who knows?
"But I'm not going to suggest a target of 340-360 is easy on the last day of any international match.
"Let's see where we are at tea, but essentially we'll be looking to keep the series at 0-0," Woolmer added ahead of the second Test at Old Trafford starting on July 27.
England began the final session on 154 for four after Kevin Pietersen and all-rounder Paul Collingwood, whose Test-best 186 was the centrepiece of England's first innings 528 for nine declared, had fallen to Afridi and Kaneria respectively.
Strauss was 74 not out and Ian Bell, one of three first innings century-makers, unbeaten on four.
But with Strauss one away from his century, Bell was run out for 28 by Pakistan captain Inzamam-ul-Haq's direct hit from mid-wicket after the England skipper hesitated after setting off.
"I was a bit of an idiot," Strauss admitted. "Being on 99 somehow manages to change pretty sane men into idiots at times and I feel very sorry for Belly."
However, the 29-year-old got the run he wanted when a single off Afridi took him to a 168-ball hundred with 11 fours.
Two balls later Geraint Jones fell to Kaneria, Strauss nicked him to slip Imran Farhat.
Earlier, England wrapped up Pakistan's first innings for 445 - a lead of 83 - after Mohammad Yousuf was last man out for a superb 202.
Yousuf's innings was the highest Test score by a Pakistan batsman at Lord's, surpassing Mohsin Khan's 200 of 1982.
It was the 31-year-old's fourth Test double hundred and second in as many matches against England after he'd made 223 at Lahore in December.
In all Yousuf, who last year converted to Islam from Christianity, batted for 10 minutes short of eight hours, facing 330 balls with one six and 26 fours after arriving at the crease with Pakistan in trouble at 28 for two.
He also matched the 2003 achievement of South Africa's Graeme Smith (277 at Edgbaston and 259 at Lord's) in making double centuries in successive Tests against England.
After play resumed Harmison hit Afridi a painful blow on the thumb, a worry for a Pakistan side already without injured pacemen Shoaib Akhtar (ankle), Rana Naved-ul-Hasan (groin) and Mohammad Asif (elbow) as well as batsmen Shoaib Malik (elbow) and Younis Khan (knee).