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The third umpire overturned the on-field decision following a review which took several minutes.
Rizwan shared a match-winning 176-run stand for the third wicket with Shafique as Pakistan romped home in 48.2 overs. This was the highest successful run-chase in the history of the World Cup.
Haris Rauf (3/32) removed Dawson (34) and Olly Stone off successive deliveries in the penultimate over before Pakistan ran out Reece Topley to claim an unlikely win.
The hosts chased down a target of 200 runs in style with Babar scoring 110.
Rizwan sustained the strain while landing awkwardly on his right leg in his effort to collect a delivery from Mohammad Hasnain during the Indian innings.
Results 1-5