The Toffees took the lead four minutes after the break on Saturday when Fellaini swept in from close range before substitute Vellios netted a second 12 minutes from time to give David Moyes' men back-to-back Premier League wins for the first time this season.
Wanderers had earlier had David Wheater sent off, the defender's 20th-minute challenge on Diniyar Bilyaletdinov earning him a straight red card - his second dismissal of the campaign.
The result moves Everton into the top half of the table, while Bolton, having suffered a 10th league defeat this term, remain in the relegation zone.
The one change to the the Trotters' team from the 2-1 defeat at West Brom last weekend saw Zat Knight come in at the back for the injured Joe Riley, with on-loan Dedryck Boyata not yet fit enough to make his return to the squad.
Toffees manager David Moyes - still without sidelined trio Jack Rodwell, Phil Neville and Sylvain Distin - also made just one alteration from his side's 2-1 win over Wolves, promoting Bilyaletdinov to the first XI in place of Royston Drenthe, who missed out due to an ankle problem.
Seamus Coleman burst forward for Everton in the third minute, but saw his shot blocked by Paul Robinson's sliding tackle, and Bolton goalkeeper Jussi Jaaskelainen palmed the ball away when Phil Jagielka got a flick to Leighton Baines' free-kick.
The visitors won a couple of corners, with Tim Cahill nodding one into the back of Wheater and the other bringing a save out of Jaaskelainen direct from Bilyaletdinov's delivery.
Toffees stopper Tim Howard was then called into action, punching clear Ivan Klasnic's strike from outside the area.
Six minutes later Wanderers were reduced to 10 men, Wheater being shown a straight red card by referee Michael Oliver after the former Middlesbrough man lunged in on Bilyaletdinov having lost possession.
Howard caught a looping effort from Chris Eagles, but Everton were looking to make their numerical advantage count, Bilyaletdinov firing over from Coleman's cut-back and the Russian then trying to feed Tim Cahill, who was crowded out before he could shoot.
A passing move almost put Bilyaletdinov through but the ball escaped him, and Jaaskelainen got down to save John Heitinga's drilled effort from Leon Osman's free-kick lay-off.
Everton continued to press, with Tim Cahill sending a diving header into Knight and Robinson getting in the way of another Coleman foray.
Bolton survived a scare shortly after the restart when Gary Cahill just beat Tim Cahill to Coleman's centre to head away, but within seconds the Toffees were in front, Marouane Fellaini sliding in to stab a cross from the left into the roof of the net.
Everton kept their foot on the pedal, with Baines whipping a dangerous ball across the box, Coleman seeing a deflected effort gathered by Jaaskelainen at the second attempt and Louis Saha shooting at the Finn from the edge of the area.
Jagielka headed a corner wide with the Toffees protesting that Kevin Davies had handled and Baines rifled an attempt over, before a rare Bolton attack saw Howard deny Klasnic with his legs.
Back came the visitors, Jaaskelainen pushing away Tim Cahill's bouncing header before blocking Saha's follow-up.
Osman was next to bring a save out of Jaaskelainen, but there was nothing the goalkeeper could do in the 78th minute as Everton finally made it 2-0.
Baines crossed the ball in, Tim Cahill stretched to knock it back across the face of goal and Vellios, who had come on for Saha, had no problem converting to wrap up the three points.