The win was a sixth in a row for Norwich as they took full advantage of the defeats suffered by top two Derby and Middlesbrough on Saturday to move to within three points of the summit.
They now appear well placed to make an immediate return to the top flight, with new boss Alex Neil having instigated a stunning run of form since taking over in January.
Ipswich, meanwhile, have now slipped below their local rivals in the table, and on this evidence face a tough battle to overhaul them.
Grabban was a central figure as the first East Anglian derby at Carrow Road for five years got off to a predictably frenetic start.
The in-form Norwich striker worked his way into space in the opposition area after just three minutes before being closed down by Tyrone Mings, then darting into the box and drawing a risky challenge from Luke Chambers, with referee Paul Tierney awarding just a corner.
Richard Chaplow was then almost punished for failing to clear his lines as Jonny Howson hooked the ball back into the danger area and Grabban nearly got on the end of a loose ball in the six-yard box.
It had nevertheless been a poor first quarter, but the Norwich fans were not complaining after 24 minutes as their side took the lead through another cracking goal from Johnson, who had scored a superb late winner at Blackburn on Tuesday night.
This time Johnson ghosted in on the left, and the midfielder was perfectly placed to crack the ball into the roof of the net from just inside the area when Grabban’s low cross wrong-footed the Ipswich defence.
Ipswich responded well to the setback and should have equalised on the half-hour mark when Daryl Murphy squared to the unmarked Freddie Sears, but John Ruddy made an excellent save to keep his side in front. Sears tested the goalkeeper again with a low shot that swerved in the strong wind yet was comfortably gathered.
It had been an evenly contested first half, with the one true moment of quality giving Norwich the edge.
Johnson went close to finding the back of his own net early in the second period after intercepting a deep Ipswich cross, while at the other end it took a well-timed challenge from Tommy Smith to prevent Grabban from getting a shot away.
The Canaries took the game by the scruff of the neck by stretching their lead in the 62nd minute. Substitute Cameron Jerome made an instant impact with a run and shot, with the ball taking a deflection off the heel of Grabban to deceive goalkeeper Bartosz Bialkowski.
The big Pole then stood up well to Jerome when the striker was put clean through. Ipswich still posed an occasional threat, with substitute Jonathan Parr only just wide of the target with a first-time snap-shot, but the hosts comfortably saw the game out.