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Wood, 34, discovered that he had bone stress in his elbow during a medical check-up after he had injured his right thigh during last month's first Test against Sri Lanka that ruled him out for the year.
He will join LSG for Rs 3 crore and this will be Joseph's first stint in the IPL.
England made 68/3 at stumps after bowling out Australia for 263.
Pacer Wood and batter Malan have been struggling for the last couple of days to get fit for the crucial knockout encounter.
Chasing a modest 146, England needed 15 off the final over but debutant Aamer Jamal (1/13) held his nerve against the experienced Moeen (51 not out) to close out the game for Pakistan.
Jos Buttler will miss the game as his wife is expecting their second child.
The England pace spearhead praised India for channelising emotion well.
Wood's absence means that Saqib Mahmood is in line for a Test debut at Headingley.
Chasing a target of 157, England romped home in 18.2 overs to take a 2-1 lead in the five-match series.
Jofra Archer (3/23) and Mark Wood (1/20) conceded just 43 runs in eight overs picking four wickets between them as India were restricted to just 124/7 in their 20 overs.
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