Ipswich took the lead after 26 minutes when Daryl Murphy got on the end of a deep cross from Carlos Edwards to head past Kuszczak.
Murphy almost doubled his personal tally for the day at the start of the second half when he came close twice in quick succession. First his shot was pushed wide and then, from the resulting corner, he headed just over.
Brighton got the equaliser with just over 10 minutes of normal time remaining courtesy of Will Buckley, who scrambled the ball over the line.
Neither side could find the winner in the closing stages, meaning that Ipswich remain in the bottom three and Brighton are usurped at the top of the league.