Everton paraded club-record signing Gylfi Sigurdsson prior to kick-off and there was further reason to celebrate when Keane headed home the opener in the 30th minute.
The match was disrupted by clashes between Hajduk supporters and stewards, with referee Ivan Kruzliak temporarily calling the players off the pitch and away from the unrest.
However, Ronald Koeman's side remained in their groove upon the restart and, while they had recorded 1-0 wins in each of their three prior competitive matches this term, there appeared little chance of the visitors holding them to just the one goal.
Gueye doubled Everton's lead before the interval and the hosts then delivered a professional second-half performance to see out a fourth successive clean sheet and take a two-goal advantage to Croatia.
Sigurdsson was presented to the home fans before kick-off and his new team-mates soon put Hajduk under pressure, with Wayne Rooney leading the line and nodding the first chance high and wide from Kevin Mirallas' corner.
Rooney teed up Ademola Lookman to tease a low centre across the face of goal, just beyond the stretching left boot of Davy Klaassen, and a series of dangerous deliveries from Cuco Martina caused the visitors further problems before Everton deservedly led.
Leighton Baines stood up a cross from a half-cleared set-piece for Keane to power a header past the stranded Dante Stipic.
Unsavoury scenes involving a section of the Hajduk supporters led to a pause in play of roughly four minutes, but Everton had a second goal before half-time.
A tight offside call had denied Rooney moments earlier, but Gueye timed his run to meet the forward's measured pass - after good work from Klaassen - and squeeze a finish inside the bottom-right corner.
Rooney, still waiting on his maiden European goal in Everton colours, was again given opportunities to get on the scoresheet either side of the break but shot too close to Stipica and then directed a header wide.
At the other end, Jordan Pickford had only previously been worked by optimistic efforts from distance, but the former Sunderland man extended himself well to save from Franck Ohandza after Hamza Barry stripped Gueye of the ball.
And Pickford saved twice more in the final 10 minutes, keeping out Ante Erceg and Hysen Memolla, to deny Hajduk an away goal and maintain Everton's control of the tie.
Key Opta stats:
- Everton have won their first four matches of a season for the first time since 1978-79, when they were managed by Gordon Lee.
- Wayne Rooney registered his first Everton assist since May 2004, when he set up Kevin Campbell for a goal against Manchester City in a 5-1 defeat on the final day of the 2003-04 Premier League season.
- Idrissa Gueye scored only his third goal in 77 appearances since moving to English football in August 2015.
- Everton have won four matches in a row without conceding a goal for the first time since September 2009.
- Jordan Pickford has kept three clean sheets in his first three Everton games, only two fewer than he managed in 35 appearances for previous club Sunderland.