John Stones, Jordan Henderson and Adam Lallana will replace Chris Smalling, Dele Alli and Daniel Sturridge.
Joe Hart, who made a deadline-day loan move to Torino from Manchester City, will keep his place in goal despite not making a league appearance this season.
The game in Trnava is Allardyce's first since taking over from Roy Hodgson.
Captain Wayne Rooney will also become England's most capped outfield player when he leads the team out on Sunday, overtaking David Beckham's record of 115 appearances.
His Manchester United team-mate Luke Shaw, who had earned his first call since breaking his leg last year, was rested from training on Saturday and subsequently removed from the squad.
Allardyce said Shaw's withdrawal was "for precautionary measures", adding "he's not injured".
The game is the Three Lions' first since the shock 2-1 defeat by Iceland that sent them out of Euro 2016 at the last-16 stage in June.
England begin their 2018 World Cup qualifying campaign against a side who held them to a 0-0 draw in the group stage of the European Championship in France.
Slovakia, ranked 24th in the world, will take belief from that draw but will also be aware that their visitors have won each of their last 12 qualifying matches for major tournaments.
England's last defeat in a qualifying game came in 2009 when Fabio Capello's England side lost 1-0 to Ukraine in Dnipropetrovsk in a 2010 World Cup qualifier.
The danger for England is that Slovakia have lost just once in 14 international matches played at home.
Allardyce said the game will be the hardest his side face in their World Cup qualifying group which also includes Scotland, Slovenia, Lithuania and Malta.
The former Sunderland and West Ham manager said he will be nervous before managing his country for the first time.
"I think in every new job, you're nervous," he said.
"But you're hoping the players will do the talking for you, that when they walk over that white line they'll deliver what we all want and that's the victory."
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