Ipswich went ahead in the first minute through Ian Westlake's lob after Kevin Pressman saved Darren Bent's shot.
David Connolly equalised 12 minutes before the interval with a low shot after James Scowcroft's flicked header skimmed past the Ipswich defence.
On 59 minutes Scowcroft knocked the ball down to Heath, who shot low into the corner, but in the last minute the Foxes defender headed past Pressman.
Ipswich manager Joe Royle:
"I think the referee is a good referee but he's got to be embarrassed when he sees it on the video. The goalkeeper's done a bar of soap and no-one's been near him. We scored a good goal and had we scored earlier we'd have gone on and won the game.
"The players are adamant no-one was near him. You don't often see our players animated around referees. We don't bother referees but they were so annoyed by that decision."
"The second half, their goal apart was so one-sided it was untrue and if we'd
have got nothing out of that game it would have been really unjust.
Leicester: Pressman, Makin, Keown, Heath, Stewart, Scowcroft,
Williams (Gudjonsson 56), Nalis, Tiatto, Connolly,
Dublin (Benjamin 80).
Subs Not Used: Gemmill, Elliott, Blake.
Goals: Connolly 34, Heath 60.
Ipswich: Price, Diallo, De Vos, Naylor, Wilnis, Miller,
Horlock, Magilton (Counago 62), Westlake, Bent (Bowditch 84),
Kuqi.
Subs Not Used: Richards, Mitchell, Supple.
Goals: Westlake 2, Heath 89 og.
Att: 22,497
Ref: A Leake (Lancashire).