Everton 2 Aston Villa 0
Wasteful finishing, good goalkeeping, bad luck and gritty defending had kept the game goalless until deep into the second half, but Radzinski's arrival changed the game.
The Canadian had come close almost immediately after his arrival, and got the crucial breakthrough when he headed home superbly after quality work from Wayne Rooney down the left.
A quick-fire second then wrapped up the points, with Gravesen beating three men before slotting home into the far corner to the delight of the Everton faithful.
Villa's Darius Vassell had missed several key chances in the Midlands derby against Birmingham, and the England striker will again be hoping that Sven Goran Eriksson does not watch a video of this game.
Although he deserves credit for being in the right place at the right time, Vassell's profligacy was illustrated early in the second half when woeful defending by Everton let the ball drop to the forward who directed his volley far too close to Nigel Martyn and allowed the Toffees keeper to make a fine reaction save.
Martyn had also denied Vassell from close range on the ground when the otherwise disappointing Lee Hendrie had done well to cut-back - but they were rare chances for a Villa side that sat back and invited pressure on to themselves.
A great tackle by Ulises de la Cruz had denied Wayne Rooney an early chance, and the teenager then missed a fine opportunity when he allowed Olof Mellberg to get back after being found by a quick free-kick.
The Toffees started to apply more pressure in the second half, and Duncan Ferguson should have done better with a free header after a fine David Unsworth cross found him unmarked - Thomas Sorensen making a straightforward save.
Thomas Hitzlsperger had a typical long range shot flash past the far post, but the game turned when Steve Watson was replaced by Radzinski.
Everton end a run of nine games without a win, and in doing so underline how quickly Aston Villa's recent good form has vanished.