Everton 1 Millwall 0: Millwall are shown no mercy by old boy Cahill

At last Everton have found a cup competition that seems to suit them. After three first-stage eliminations out of three this season, they booked a fourth-round home FA Cup tie against Chelsea thanks to Tim Cahill.

Only 18 minutes remained when the Australian midfielder struck, his muted celebrations underlining the affection he still holds for his old club.

But the goal, which might also have gone in off the challenging Alan Dunne, was long overdue and well deserved as Everton also hit the post twice.

Cahill's last game for Millwall prior to a £1.5m move north was the 2004 FA Cup final, since when he has had one prolific campaign and one thus far less so - at least until his current purple patch of four goals in five appearances.

Unusually, he started last night almost up alongside James Beattie but switched back to his orthodox role long before his fifth goal of the season secured a fourth-round date described by Alan Irvine as: "Fantastic. It's absolutely brilliant and it's what you come into football for." The Goodison No 2 added: "We deserved to win and should have won down there. It's the same story... we missed chances.

"It's great that we're creating chances but we have to be more clinical. We don't have much money to spend, so we won't be signing a 30-goal striker this month. Whether we sign anyone depends on whether we can find the right man."

The green shoots of recovery are again peeping through Everton's depressing season, with three successive Premiership victories woven around the draw secured late on by Leon Osman at the New Den 12 days ago.

David Moyes' side often dominated opponents one place off the bottom of the Coca-Cola Championship, although the absence of an on-target attempt until the 52nd minute was a worry.

The approach work was usually impressive but any number of efforts flew high or wide. Andy Marshall was not troubled at all in the first half and then, having acrobatically tipped over Cahill's header to a superb centre by Osman, was reprieved by a bad miss from Kevin Kilbane, who scooped too high when Osman's shot thudded out off his chest.

Kilbane was left bemused shortly afterwards when his header to Mikel Arteta's corner bounced off the post with the keeper motionless.

The pressure finally told just as Everton were fearing extra time two and a half days before the Premiership visit of Arsenal.

Cahill lunged with Dunne at a square pass into the area by Beattie and, with his left foot and possibly his marker's help, sent the ball arcing high and wide of Marshall.

Millwall, having harvested eight points from four matches under the fledgling managership of David Tuttle before Saturday's defeat at Preston, pushed forward when they could.

They were encouraged by edginess in a home defence already weakened by the departure of Joseph Yobo to the African Nations' Cup and further weakened from just past the half-hour mark when the captain, David Weir, went off with a cut at the top of his nose.

All they could really muster by way of a goal threat, though, was a promising shot by Tony Craig that hit his captain, Dave Livermore, and a late long-range effort from the latter that Nigel Martyn dived to save.

Tuttle commented: "I'm pleased we contained Everton and looked reasonably solid but our lack of penetration emphasised why I'm trying to sign a striker or two during the transfer window."

Everton (4-4-1-1): Martyn; Hibbert, Weir (Kroldrup, 33), Ferrari, Valente; Osman, Arteta, Neville, Kilbane; Cahill; Beattie. Substitutes not used: Wright (gk), Naysmith, McFadden, Anichebe.

Millwall (3-5-2): Marshall; Robinson, Lawrence, Whitbread; Dunne, Livermore, Morris (Cogan, 77), Elliott, Craig (Braniff, 88); May, Williams. Substitutes not used: Healy, Pooley, Hendry.

Referee: R Styles (Hampshire).