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Andy Marshall made a good early save to deny Adam Bolder, before Darren Ambrose failed to control a ball from Jim Magilton which would have seen him clear.
Mart Poom kept out a Matt Holland volley, before the Estonian pulled of a flying stop to deny Magilton.
Darren Bent failed to connect with a dangerous Pablo Counago cross in a good position and then Ipswich lost Fabian Wilnis to injury after just 21 minutes, Jermaine Wright replacing him.
Magilton curled a free-kick over from a central area before Derby took the lead with a controversial strike.
Wright was adjudged to have fouled debutant Izale McLeod wide on the left, and from Rob Lee's curling free-kick, Carbonari stole in behind Herman Hreidarsson to beat Marshall from close range.
Marshall was booked for disputing the free kick decision, one of five first-half cautions dished out by fussy referee Keith Hill.
McLeod had what would have been a dream debut goal chalked off for offside from Malcolm Christie's pass, before a low Counago effort was scrambled away from sub Finidi George by Paul Boertien.
Town should have levelled on the hour, but Bent sidefooted over an open net from four yards after Wright's low cross teed him up.
Bent was guilty of an equally bad miss with his header from a George cross late on, but by then Christie could have sealed it with a solo run which was only halted by a fine Marshall save.
Despite almost six minutes of added time, Town showed a worrying lack of creativity and slipped ever nearer to the bottom of the table.