The two sides were equal for much of the first hour before the hosts' captain Martin Skrtel was rightly sent off for a second yellow card offence.
But despite their advantage, England couldn't force the breakthrough as Lallana went closest with a number of chances, including striking the post after a clever turn on the edge of the area.
The Three Lions - having endured a goalless draw with Slovakia at Euro 2016 only in June - looked like they would be frustrated again with Allardyce on course to become the first England boss since Sir Bobby Robson in 1982 not to win his first game.
But in the dying moments, attacking midfielder Lallana conjured up a winner to open their 2018 World Cup qualifying campaign with a win, saving his new manager's blushes in the process.
Few surprises from Big Sam's first England eleven
Former Sunderland boss Allardyce made just three changes from the England side that crashed out of the Euros to Iceland in their last fixture three months ago.
Manchester City centre-back John Stones and Liverpool duo Jordan Henderson and Lallana came in at the expense of Chris Smalling, Dele Alli and Daniel Sturridge in his first-ever eleven.
Of the eight players to keep their places, goalkeeper Joe Hart, who made a deadline-day season-long loan move to Torino from City, remained in goal despite having yet to make an appearance at club level this season.
Having confirmed that Wayne Rooney would remain as captain, the Manchester United forward became England's most capped outfielder in leading the team out - overtaking David Beckham's previous record of 115 appearances.
Slovakia head coach Jan Kozak also made a handful of changes, with five different players making the line-up from the team he fielded as they fell to defeat to Germany in the last 16 of the Euros.
Centre-back and captain Skrtel, who left Liverpool for Fenerbahce in the summer, kept his place as did Napoli midfielder Marek Hamsik - of their most notable names.
The two sides only last met in the summer in the final group stage game of the Euros, with former boss Roy Hodgson's wholesale changes that night costing England dear as the two sides played out a drab goalless draw in Saint-Etienne.
And unsurprisingly, it took time for Allardyce's charges to settle in to their stride at the City Arena - as they set up in a fluid formation that blended between a 4-2-3-1 and a 4-3-3, with Rooney often dropping deep into central midfield.
Dubbed their most difficult qualifier by Allardyce in the build-up, the 61-year-old admitting that he would be happy to return home with a point, Slovakia's defensive rigidity and resilience kept the visitors at bay for much of the first-half.
It wasn't until the 21st minute that England had their first half-chance, Kyle Walker rolling a cross into the path of Harry Kane in at the near post only for his Tottenham Hotspur teammate to miss the ball altogether from a promising position.
Just minutes later, Raheem Sterling - drifting into space between the Slovakian back-line from the left side - couldn't quite bring Henderson's ball through the lines under control as England continued to probe for openings.
Two sides even in tightly-contested first 45 minutes
An extremely narrow England side failed to provide sufficient support for lone frontman Kane and Allardyce regularly attempted minor tweaks - Sterling and Lallana occasionally swapping flanks - in an attempt to get more bodies in and around the No.9.
But the changes were to no avail and England were fortunate not to fall behind on 37 minutes, with Danny Rose guilty of taking too much time on the ball inside his own area.
The left-back was dispossessed by Michal Duris and his fizzed ball across the penalty box evaded the lunging Dusan Svento by just a matter of inches.
Almost immediately up the other end, Sterling dragged agonisingly wide from inside the area after Kane's pull-back in search of the winger had initially been intercepted.
A mix-up between two Slovakia defenders allowed the ball to inadvertently fall for Sterling, but after taking a touch to bring it under control, his low right-footed drive zipped wide of the far post.
Just before the break, Skrtel was fortunate not to see a second yellow card for bringing down Kane from behind having earlier gone into the book for elbowing the same player.
England came out after half-time with plenty of early purpose, Sterling seeing a shot deflected just wide, but soon returned to their ponderous and pedestrian possession play.
Skrtel's red card gives England the advantage
They retained the same shape, which did little to disturb Slovakia's well-drilled defence although Lallana and Sterling noticeably remained wider rather than drifting inside.
But the Three Lions were dealt the initiative when Skrtel was shown a second yellow before the hour mark, punished for accumulating too many fouls and a vicious stamp down on Kane's leg, which arguably could have been a straight red were he not already booked.
Allardyce sought to take advantage, and inject some extra tempo into England's play, with the introduction of Dele Alli after 63 minutes - the Spurs midfielder taking the place of Henderson in the centre.
But just before his withdrawal, Henderson could and should have had an assist for an intelligent run and lay-off for Lallana - who could only curl wide in searching for the top corner.
England were unquestionably the team on top with the numerical advantage, but still struggled to turn their dominance into goals as they enjoyed a spell of sustained pressure.
The final ball continued to let them down with Lallana close to latching on to Alli's through ball but lacking a yard of extra pace to beat the goalkeeper to it.
Allardyce was bemoaning the decision-making of the referee soon after, when Serbian official Milorad Mazic ended advantage to award a free-kick with Sterling having been fed through on goal just as he blew his whistle.
The England boss looked to Theo Walcott off the bench to help inspire his side to a winner in final 20 minutes, but still Slovakia battled to deny them any clear chances.
Lallana pops up with first England goal late on
Lallana came the closest of anyone in a red shirt with two shots in quick succession in the 75th minute, one striking the inside of the post but ricocheting out before Matus Kozacik was forced to beat away another attempt.
A goal seemed to be on its way and Allardyce's final roll of the dice saw Daniel Sturridge brought on to replace Kane up top, giving the away side extra pace to try and get in behind.
A move involving both substitutes Sturridge and Walcott saw the latter find the back of the net late on, only to be denied by the linesman's flag for a tight offside call.
Then, with seconds remaining of added time, Lallana - on his 27th cap - jinked into space and squirmed the winning goal underneath Kozacik with the very last kick of the game to secure a seal a 13th straight qualifying win for England and jubilantly kickstart the Allardyce era.