Having also cruised to the next round of the EFL Cup with a comfortable 4-0 triumph against League Two's Yeovil Town, it means Koeman is yet to taste competitive defeat as Everton manager so far.
But they'll face a tough test to maintain their unbeaten start as Stoke hope to bounce back from a chastening 4-1 reversal to Manchester City a week ago, which followed on from a 1-1 draw at Middlesbrough.
But Mark Hughes' men might take hope from the fact they have won three of their last four clashes with Everton at Goodison Park, including the last two.
Stoke will also be pleased with the manner in which they thrashed Stevenage in the cup in mid-week, as 35-year-old frontman Peter Crouch celebrated a hattrick in a 4-0 win.
They have yet to truly click into gear this term, despite only making one high-profile signing in the £13.5 million addition of midfielder Joe Allen from Liverpool.
However it is perhaps too easy to read too much into their defeat to City, having succumbed to a harsh penalty decision, before conceding two late goals to distort the scoreline in the visitors' favour.
As such, Koeman will be well aware of the threats they do possess, as he too looks to ensure what could be a new-look Everton side against Stoke go into the first international break in good shape.
The Dutchman has invested heavily into the squad, most notably with the £25 million capture of Yannick Bolasie as well as a £12 million move for Ashley Williams and Idrissa Gueye's arrival for £7.1 million.
While Gueye has slotted in seamlessly, Williams and Bolasie are still yet to make their league debuts after a lack of match fitness and Koeman will hope the pair can only add to Everton's positive start.
Their signings have lessened the blow of losing centre-back John Stones to Manchester City for an initial £47.5 million, while they have kept hold of key players such as Romelu Lukaku and Ross Barkley.
Lukaku has yet to feature this season after injury, but getting him fit - and performing to the levels of last season when he hit a career-best 18 league goals - will be top of Koeman's priorities.
Stoke, too, will look towards previous season's achievements as they hope to maintain back-to-back ninth-placed finishes. Though they could go into the international break without a win, Hughes and co. will be confident they have what it takes to get off the mark sooner rather than later.
The wealth of talent at the Welsh manager's disposal - including Xherdan Shaqiri, Bojan and Giannelli Imbula - should see Stoke challenge for a top-half spot once again.
But they may need more additions before the closure of the transfer window, with a striker and a centre-back thought to be high on Hughes' list.
Regardless, they have the threats to hurt hosts Everton - who will need to perform well in order to continue pace with the other seven teams yet to lose this season.
Summer signings Bolasie and Williams could be handed their first starts for Everton after making their full debuts against Yeovil, while Lukaku could make a first start.
The trio all came off the bench for their first competitive appearances of the season at West Brom and are believed to have achieved sufficient match fitness to be able to start.
Right-back Seamus Coleman will miss out with a troubling ankle injury, while James McCarthy, Darron Gibson and Tom Cleverley are all doubts with unspecified knocks.
Long-term absentee Muhamed Besic (knee) and young defenders Tyias Browning (knee) and Matthew Pennington (hamstring) also remain out.
Shaqiri could return for Stoke from a slight calf problem that has kept him out of the last two games since scoring in their league opener, while one of Crouch, Jonathan Walters or Mame Biram Diouf is likely to lead the line.
Goalkeeper Jack Butland remains out with an ankle problem, but could be back for the Potters' first game after the international break on September 10.
Glen Johnson (thigh), Stephen Ireland (broken leg) and Ibrahim Afellay (knee) are unavailable, meaning they could make few changes from the team that started against City.
At Goodison Park: Everton - wins, Draws - , Stoke City - wins.
At the Bet365 Stadium: Stoke City - wins, Draws - , Everton - wins.
Overall: Everton - 61 wins, Draws - 33, Stoke City - 35 wins. (as per 11v11.com)
Last meeting: Stoke City 0-3 Everton (Lukaku, Coleman, Lennon), 2nd February 2016, Barclays Premier League.
Form in Premier League: DW
Form in all competitions: DWW
Form in Premier League: DL
Form in all competitions: DLW
Stoke have won on their last two trips to Goodison Park and have come away with three points from three of their last four away games against Everton.
The last meeting between the sides finished firmly in Everton's favour, however, as they cruised to a 3-0 away win in February.
Three points would give Everton their highest total of points after three games since the 2006 season.
Romelu Lukaku has gone 10 Premier League games without a goal, stretching back to March, his longest-ever drought in Englishf ootball.
Stoke City have failed to win either of their first two games for the sixth time in the last seven seasons.
The Potters have conceded 41 goals in 2016 - more than any other side in the division - and haven't kept a clean sheet in 12 league games.
Mark Hughes' men have picked up just two wins from their last 10 away games, losing six.
Michael Oliver is the nominated referee for Saturday's clash.
The 31-year-old, who was promoted to the promoted to the Select Group of Referees in 2010, remains the youngest-ever official to take charge of a top-flight game.
Oliver - who was the man in the middle for last season's League Cup final - has already handed out 13 yellow cards across two games so far this season, but no reds.
He oversaw 26 Premier League games last season, handing out 92 yellows and five red cards in total.
In the games he refereed involving Everton, the Merseyside outfit won twice, drew once and lost once while he only took charge of one Stoke game - which they lost.