England were 37 for two in their second innings at lunch on the third day - still needing a further 142 runs to make Pakistan bat again.
"We've had two good days and we'll enjoy them, but we've got a hell of a lot of work still to do", he told Sky Sports.
"With Dom Bess, it was like Joe Root walked back out, with his back foot punches and cover drives, " Buttler added.
"Pakistan, however, should still complete a deserved victory on Sunday".
Alastair Cook, in his 153rd consecutive Test which equalled Australian Allan Border's all-time record for successive appearances at this level, was the only England batsman to play an innings of substance with 70.
The tourists scored 300 runs in Friday's play, going from an overnight 50 for one to 350 for eight at stumps on the second day.
Wood made the breakthrough when Sohail edged an outswinger to wicketkeeper Jonny Bairstow to depart for 39.
Ben Stokes clipped his first ball through square leg for four and followed up with another crisp boundary but, on nine, he tried to hit Shadab through the leg-side and was well caught at short mid-wicket.
Babar was on 68 at the time and after assessment in the middle he retired hurt.
A four by Buttler, selected on the basis of his form in the Twenty20 Indian Premier League, saw him to an 89-ball fifty, with the 20-year-old Bess - in his first match at Lord's - receiving huge cheers when he reached the landmark in 93 balls.
He played some polished attacking strokes and his fifty was greeted by warm applause from the packed crowd as he raised his bat to all corners of the ground.
England, bowled out for 184, toiled on another overcast day at the home of cricket, dropping several catches in a sloppy performance which did little to lift morale following their heavy Ashes defeat and series lost in New Zealand.
But at least Buttler, who finished on 66 not out, and Bess (55 not out) justified their selections and gave their team a bit of confidence ahead of a potentially even more hard five-test series against top-ranked India.
Earlier, Pakistan were dismissed for 363 - a first-innings lead of 179. Checks revealed a fracture and he was ruled out of the remainder of this test and for the second and final test in Leeds. England were a mere 56 runs ahead with four wickets standing and the new ball on the horizon, but at least they were still in the game.