Marouane Fellaini scored Everton's fifth goal of the night in a 5-1 win at Cheltenham Town that completed a one-sided FA Cup third round win at Whaddon Road.
Nikica Jelavic, Leighton Baines, Leon Osman and Seamus Coleman were also on target before Fellaini netted in the final minute to secure David Moyes's side a trip to Bolton or Sunderland in the fourth round at the end of the month.
Russell Penn scored for League Two Cheltenham after Everton had gone 3-0 ahead early in the second half, but the home side struggled to cope with the quality of the visitors, who are looking to go one step further having lost to Liverpool in the semi-finals of last year's competition.
Everton have now conceded a goal in 18 consecutive matches in all competitions, but they will be content with this outing. Moyes was rewarded for resisting the temptation to tinker with his starting line-up as the Premier League team responded with a display of real professionalism.
Everton made two changes from their win at Newcastle as Coleman and Bryan Oviedo replaced John Heitinga and Steven Naismith before a crowd of 6,891.
Everton moved ahead with their first attacking move of the evening as Croatia forward Jelavic tucked the ball into the gaping net after a Fellaini shot had come back off a post from a lovely flick by Oviedo.
They moved two goals to the good when Alan Bennett was harshly deemed to have been climbing on the shoulders of the grounded Fellaini.
Baines composed himself to sweep the ball to the right of the home goalkeeper Scott Brown, who guessed right but could not cope with the pace of the finish.
With Everton keeping the ball almost at will against the game but sporadic attacks of Cheltenham, Osman helped himself to his side's third after Baines had picked him out with a lovely cutback from a rampaging run.
Everton dozed off as Penn finished Kaid Mohamed's ball from close range, but another fine counter attacking move by Everton ended with Victor Anichebe picking out Coleman to produce a superb chipped finish.
Anichebe was again involved as he played provider for a second time to find Fellaini, who finished with some aplomb with Cheltenham clearly tiring in the death throes of an entertaining if one-sided contest.
Cheltenham had conceded over four goals in an FA Cup game at home for the first time since November 1968 when lost 0-4 to Watford, but they will be perhaps be relieved not to have shipped more to a visiting side who showed why they are fifth in the Premier League.
MAN OF THE MATCH
Leighton Baines (Everton): Superb showing from the full-back. Spent most of the night on the attack.
Cheltenham: Brown 6, Jombati 6, Bennett 6, Elliott 6, Jones 6, Pack 6, Carter 6, McGlashan 6, Penn 7, Mohamed 6, Goulding 6. Subs: Deering 6, Duffy 6, Taylor 6.
Everton: Howard 7, Coleman 8, Baines 8, Distin 8, Jagielka 8, Oviedo 7, Neville 7, Osman 8, Fellaini 8, Anichebe 7, Jelavic 7. Subs: Gueye 6, Hitzlsperger 6, Naismith 6.