It was a dismal start for Everton manager David Moyes and chairman Bill Kenwright, who have come under fire for their failure to land a single new player this summer.
In sharp contrast, Ince's new charges delivered a spirited performance that answered claims of dressing room unrest at Ewood Park.
In front of the watching Sir Alex Ferguson, Ince's side exerted almost total domination in the first 45 minutes, to the increasing agitation of the home crowd.
Everton keeper Tim Howard saved well from Morten Gamst Pedersen and Santa Cruz - a continuing transfer target for Manchester City and - twice headed off target from good positions following dangerous crosses by Steven Reid.
Ince was delighted with victory in his first match in charge
Blackburn deserved to be ahead, and they were finally in front after 22 minutes thanks to a brilliant finish from Dunn.
He cut inside Phil Jagielka before curling a superb strike beyond the reach of Howard for the first Premier League goal of Ince's reign.
Everton's attacking efforts had been virtually non-existent, but they were back in business with almost the last kick of the opening half.
Arteta swerved in a stunning free-kick from a tight angle, but questions must be asked of the positioning of England keeper Robinson on his Blackburn debut.
Blackburn continued to look the more composed side, and it took a vital interception from Joseph Yobo to deny Jason Roberts after he weaved his way into the area.
But Blackburn had not regained the supremacy they enjoyed in the first half, and they fell behind after 64 minutes as Everton carved out an excellent second goal.
Yakubu set Arteta free on the left flank, then raced to the far post to get on the end of a perfect cross and head beyond Robinson.
Everton's lead lasted only two minutes when a moment of defensive slackness gifted Blackburn an equaliser.
'Moyes thought the winning goal was offside
Stephen Warnock's long ball dropped behind Joleon Lescott, and Santa Cruz slid a simple finish past Howard as Everton appealed in vain for offside.
Everton brought on 16-year-old Jose Baxter as a late susbstitute, and he almost claimed the points with a minute to go, sending a stretching header off target from Jagielka's cross.
But with only seconds remaining Blackburn took the points as Ooijer pounced after Nelsen headed Warnock's free-kick against the woodwork.
Everton manager David Moyes: "Our defending at times was rubbish. We had experienced players, internationals, who were not at the races.
"We gave away a daft free-kick for their winner, and even then Ryan Nelsen was offside when the ball came into the box.
"It was hard for the linesman to see that, but Nelsen had moved from an offside position to head against the bar.
"On the balance, Blackburn probably deserved the win, but we had enough chances and once we were ahead we should have had defenders out there who could have hung on to the victory."
Blackburn manager Paul Ince: "We have won at Everton and not many teams do that apart from the top four. It was a great achievement.
"The whole performance was fantastic, to come from behind twice and win was outstanding. I would have settled for a point at one stage."
"We want to keep our best players, be it Roque Santa Cruz, Stephen Warnock or Benni McCarthy and any one of a dozen players.
"Let's dispel all this. No-one is leaving the club, we have had a really good start in this game."
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