Aidy White and Tom Lees recovered from niggling injuries to take their place in the back four, but Billy Paynter was not involved after after missing the past three weeks with a thigh injury.
Manchester United will rekindle their rivalry with Leeds with a trip to Elland Road in the third round of the Carling Cup.
Leeds last met Manchester United in the third round of the FA Cup in January 2010, winning 1-0 at Old Trafford with a Jermaine Beckford goal.
At Portman Road, United took the lead on 34 minutes, Ross McCormack on target for the visitors, who had dominated.
United were reduced to 10 men on 49 minutes when White was sent off for a professional foul on Jay Emmanuel-Thomas.
Ipswich made their numerical advantage count as Jason Scotland equalised with 13 minutes remaining, firing home from 25 yards.
Andrews then struck to win it for Town. The former Blackburn midfielder’s late strike brought great cheer to the Ipswich fans, who had seen their side lose their previous four matches, including 7-1 and 5-2 losses to Peterborough and Southampton.
However, it was cruel on Leeds, who had performed well in the first half and looked in control of the game until White’s sending off in the 48th minute for fouling Emmanuel-Thomas as he bore down on goal.
Leeds boss Simon Grayson said: “First half we were outstanding against a team that has been put together with a lot of money and internationals and they created nothing.
“They never created a chance against us. We could have been two or three up and the game would have been put to bed.
“Ross McCormack’s header hit David Stockdale and he knows nothing about it. It wasn’t a worldy save, I could have accepted that, but we could have had the game won.
“When the home crowd boo their team off you know you have done a good job.”
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