The hosts needed to come from behind against their closest Group F challengers after Stanislav Lobotka opened the scoring inside three minutes, but the two changes to the starting lineup from the win over Malta - Eric Dier and Marcus Rashford - got the goals to fire England five points clear at the top of the standings.
The Three Lions now need just two points from their final two games at home to Slovenia and away to Lithuania to guarantee top spot in the group and a place at next summer's tournament in Russia.
Gareth Southgate was able to welcome Dier back from suspension while Rashford was handed his first Wembley start, but the latter's match got off to a nightmare start when his mistake led to Slovakia opening the scoring inside three minutes.
Rashford was caught in possession by Lobotka inside his own half, and the Slovakian midfielder then played a one-two before taking the ball down inside the area and sweeping his finish past a statuesque Joe Hart.
It was the first goal by a visiting player in a competitive match at Wembley since October 2013 and the fastest against England in a competitive game since November 1993, while Lobotka also wrote his name into the history books as the first ever Slovakian to score against England in an international match at Wembley.
Rashford immediately looked to atone for his error by driving forward at the other end and forcing a save from Martin Dubravka, but the Slovakian keeper was rarely tested as England looked for a response to the early setback.
Dele Alli and new Liverpool midfielder Alex Oxlade-Chamberlain squandered decent shooting opportunities from outside the area, before Alli got on the end of a cross from the right only to put a tame header straight at Dubravka.
Rashford wasted another sight of goal shortly after the half-hour mark, but the youngster did go some way to making amends for his earlier error by providing the corner from which England levelled things up with eight minutes to go until half time.
Dier's dart to the near post caught Slovakia's defence sleeping, and the Tottenham Hotspur midfielder managed to hook his foot around the ball to sweep the low delivery into the top corner for his third international goal.
The hosts quickly set about looking for a second goal before the interval, but Dubravka kept Slovakia ahead going into the break with saves to deny Ryan Bertrand - who only got his first England goal against Malta on Friday - and Rashford.
It was a better end to the half from England, and they picked up where they left off after the break as Alli combined with Rashford before seeing his shot blocked, while Jordan Henderson scuffed the rebound wide.
Oxlade-Chamberlain then forced another stop from Dubravka as England continued their bright start to the half, although they needed Hart to keep the scores level 10 minutes after the restart when Adam Nemec took down Marek Hamsik's cross before firing a volley towards goal.
Martin Skrtel then threatened from the resulting corner, but England were soon back on the attack and Alli could not make the most of two sights of goal in quick succession - first failing to react in time after Harry Kane's powerful cross was parried out and then firing a long-range strike well off target.
The hosts did take the lead for the first time just before the hour mark, though, as Rashford fully redeemed himself with an accurate finish into the bottom corner from 25 yards.
It capped a much-improved performance in the second half from Southgate's side, but after Alli fluffed another sight of goal Slovakia began to see more of the ball again, and Tomas Hubocan should have done better when he fired a chance over the bar after a free kick was only cleared as far as the edge of the box.
England did a good job of seeing the game out in the closing stages, though, and it was the hosts who came closest to adding another goal when substitute Raheem Sterling drew a save from the keeper moments after coming off the bench.
Rashford's winner proved to be enough to send England five points clear, though, as the Three Lions chalked up a 13th consecutive competitive win at Wembley, stretching their unbeaten streak in qualifiers to 37.
Slovakia, meanwhile, are now just one point clear of Slovenia and Scotland, the latter of whom they face at Hampden Park in their next outing for what could be a straight shootout for second place in the group.