Surrey opener Roy’s quickfire 65 fired England towards an initial victory target of 261 after Pakistanhad made 260 for six, captain Azhar Ali top-scoring with 82.
But following a third and final rain interruption of this day/night fixture at 9:37 pm local time (2037 GMT), the umpires eventually called the game off with England 194 for three.
Eoin Morgan, the England captain, was 33 not out and Ben Stokes 15 not out.
Roy and Joe Root (61) took England to the brink of victory with a second-wicket stand of 89 in 14 overs.
The 26-year-old Roy sparked England’s chase with three fours in four balls off Umar Gul.
But there was a worrying moment when Roy, on 20, needed several minutes’ on-field treatment for a dizzy spell.
“It was a case of not having enough sugar on me apparently, that’s it plain and simple,” Roy told reporters.
“I had a headache, felt a bit dizzy, they got the physio and the doctor on, they told me to get some sugar on board.
“I was able to get my bearings, settle and re-set myself and just go again,” he added.
Meanwhile opening partner Alex Hales, who managed just 145 runs during the preceding 2-2 drawn Test series against Pakistan, again fell cheaply.
Hales was out for seven when he guided Gul straight to Mohammad Hafeez at slip.
– More Amir misfortune –
Mohammad Amir had five catches dropped off his bowling during the Test series and the Pakistanpaceman’s bad luck continued Wednesday.
Roy, on 24, skied left-arm quick Amir high on the legside.
The ball appeared to be heading straight to Gul at square leg.
However, wicket-keeper Sarfraz Ahmed called for the catch, only to make insufficient ground and drop the chance despite getting both gloves to the ball.
Pakistan may have recently climbed to the top of the Test rankings, but this was the kind of fielding error which had helped leave them languishing in ninth place in the equivalent ODI standings.
Roy swept left-arm spinner Imad Wasim for the first six of the match before completing a 43-ball fifty also featuring five fours.
But his innings ended when a lofted drive off spinner Mohammad Nawaz was brilliantly caught just inside the boundary by Babar, running round from long-off.
Root, England’s star batsman in all three formats, was untroubled while hitting six fours in 72 balls.
But when Morgan dropped the ball into the offside and set off, Root failed to beat Azhar’s direct hit.
Earlier, Azhar won the toss and led from the front in an innings where Ahmed made 55 and Azam was unluckily lbw for 40.
Pakistan were going well at 173 for three after 35 overs but then lost Azhar and the skipper felt his side had fallen short with the bat.
“300 obviously, we always look to score 300, but definitely 280-290.
“One of your top order has to score and bat long, triple figures would be nice and you always build a total when the top order perform,” said the opener.
Azhar, however, was twice dropped on nine.
Hales floored a relatively routine chance in the gully before wicket-keeper Jos Buttler could not hold a more difficult left-handed catch down the legside.
Babar was given out lbw to leg-spinner Adil Rashid by Australian umpire Simon Fry, even though replays clearly showed an inside edge.
But by that stage Pakistan had used up their lone review of the innings.
Azhar pressed on to a relatively sedate 84-ball fifty before upping his tempo.
But he fell when top-edging a slog-sweep off Rashid (two for 51) to Moeen Ali at short third man.
The five-match series continues at Lord’s on Saturday.