The home side created the better chances throughout the match, hitting the woodwork twice on their way to maintaining their unbeaten record after three games in Group F.
England, without captain Wayne Rooney after he was dropped to the bench, began the match quickly and created their first sight of goal inside two minutes when Daniel Sturridge curled a snapshot wide of the far post.
The Liverpool striker scored England's opener during their 2-0 victory over Malta on Saturday, but he failed to make the most of another good chance moments later when his heavy touch from Dele Alli's incisive pass allowed Jan Oblak to push the ball behind.
Sturridge saw another effort fly over from range before the 10-minute mark, but England soon found themselves fortunate to still be level when Slovenia missed a trio of chances in quick succession.
Eric Dier, starting in place of Rooney, played a blind pass back towards Joe Hart which was cut out by Roman Bezjak, who found himself clean through on goal. However, Hart was quick off his line to meet the forward on the edge of the box and make a vital save.
The danger had not passed, though, and moments later Hart needed the woodwork to come to his rescue as Jasmin Kurtic's curling effort came back off the post before Bezjak skewed the rebound wide.
England, who fell behind in both Euro 2016 qualifying matches against Slovenia, were left to breathe a sigh of relief, and the match soon entered a lull with both sides unable to make serious headway in the attacking third.
Theo Walcott tried a long-range effort midway through the half that flew off target, while Gary Cahill, making his 50th appearance for his country, saw a goalbound header strike the back of an unwitting Kurtic before it could trouble Oblak just four minutes before the interval.
The visitors were unable to muster a single shot on target in the opening 45 minutes, though, and then found themselves once again lucky to still be level as Slovenia made a flying start to the second half.
Within seconds of kickoff Josip Ilicic got in behind the defence with a clever one-two inside the box, but with only Hart to beat the playmaker tried to pick out a teammate in the middle and only found Walcott, who hacked it away.
The resulting corner was bundled goalwards by Cesar to force Hart into a smart save from close range, but it was from a second corner that the Torino loanee produced his standout moment when he spectacularly clawed a Kurtic header against the bar and out to keep his side hanging on.
Whereas Hart had single-handedly rescued his side on a number of occasions, it took until the 56th minute for Oblak to make his first save of the contest when he comfortably gathered Alli's effort from outside the area.
Slovenia were soon back on the attack, though, and shortly before the hour mark Ilicic won the ball inside the England half before firing a low strike narrowly wide of the target from range.
Andros Townsend drew a rare stop from Oblak moments after coming on as a sub, but it was Hart at the opposite end who was the busier of the two keepers, and he was once again called upon to rescue his side when he denied Ilicic after stand-in captain Jordan Henderson had seen his underhit back-pass picked off.
England finally began to pose more of an attacking threat in the final 15 minutes after Rooney had been called off the bench, but Danny Rose, Sturridge and Rooney himself all wasted decent openings within three minutes of each other.
Oblak's most taxing save of the night arrived with eight minutes remaining when he turned Jesse Lingard's powerful 25-yard drive past the post, although it was a stop the Atletico Madrid keeper would have expected to make.
The closing stages of the match were disrupted by a number of fouls and a flurry of late cards as Slovenia held on for a valuable point, ending England's 14-match winning streak in qualifying matches.
Despite the dropped points, England remain top of Group F, while Slovenia - now the only other unbeaten side in the group - move above Scotland into second place.