The Colombia winger, who scored in last week’s League Cup win against Barnsley, fired Leeds level in the 55th minute after Anthony Gordon had given Everton a half-time lead.
A point apiece kept Leeds in fifth place and was enough to lift Everton out of the bottom three, although Frank Lampard’s side remain winless after their first five Premier League matches.
Everton weathered the inevitable early storm without conceding any clear-cut chances.
Leeds’ crisp and accurate build-up play petered out on the edge of Everton’s penalty area and the visitors snatched the lead against the run of play with their first meaningful attack in the 17th minute.
Alex Iwobi picked out Gordon’s darting run with an incisive through-ball, which caught the Leeds defence flat-footed, and the latter raced in to fire a low shot under goalkeeper Illan Meslier.
It was the Everton winger’s second goal in successive matches and credit to the 21-year-old’s temperament amid intense speculation over a possible move to Chelsea.
Jack Harrison’s excellent ball across the face of goal just evaded Rodrigo, who headed Brenden Aaronson’s cross from the opposite wing soon after off target, as Leeds pressed forward again.
Rodrigo, and Leeds, were then dealt a blow in the 30th minute when the Spaniard was withdrawn after appearing to injure his shoulder in a challenge with Everton goalkeeper Jordan Pickford.
Joe Gelhardt replaced Rodrigo despite the return to the bench of Patrick Bamford after a groin injury and Leeds continued to chase an equaliser.
For all Leeds’ pressure, Pickford did not have a save to make before half-time, but the England keeper was forced into action within 60 seconds of the restart.
He turned away Aaronson’s curling goal-bound shot and comfortably held Harrison’s low effort as Leeds swarmed forward.
But Pickford was left helpless in the 57th minute when Leeds drew level.
Gelhardt set Aaronson scampering goal-wards and his pass wide found Sinisterra, who fired a low left-footed shot inside a motionless Pickford’s left-hand post.
Tempers then boiled over, on the pitch and in the dug-outs. Gordon and Kristensen leant in with their foreheads and the subsequent melee was mirrored on the touchline after words had been exchanged between rival coaching staff.
Everton thought they had regained the lead through Demarai Gray’s angled shot, but VAR ruled he had been just offside.
Gelhardt flashed a shot wide at the other end and was denied by Pickford’s boot with another effort before being replaced by Bamford.
Both sides went in search of a winner. Leeds peppered Everton’s penalty area and Nathan Patterson’s late shot was saved by Meslier as the points were shared.