Adebayo, a loan signing from Sky Bet Championship club Fulham this week, shone throughout this tie, with his technique and strength too much for last season's semi-finalists Bognor to handle at times.
Slough joint manager Neil Baker was delighted with the win and the impact the 18-year-old made.
"We had a well-deserved half time lead and some of the football we played was exceptional, with three great finishes. We are delighted to have Adebayo, we thought he worked hard."
"On another day we could have had a few more goals, but Bognor are a very good side, we knew they wouldn't give up. We rode our luck at the end and finished it off with the fourth."
This tie promised to entertain, both clubs having made strong starts in the Southern and Isthmian Premier divisions. As early as the second minute, a Bognor corner caused problems for Slough and a header was blocked in the goalmouth by Mark Scott at point blank range.
Adebayo had already gone close before he opened the scoring on 12 minutes. Dobson swung in a trademark free kick which the striker couldn’t connect cleanly with, and the ball deflected onto the crossbar leading to a goalmouth scramble which eventually ended in the danger being cleared.
But when Adebayo did find the target after a long throw was knocked down to him, it was a crisp, powerful finish from 16 yards which gave Dan Lincoln no chance.
Slough took hold of the game and doubled their lead on 22 minutes, this time a fine individual goal from top scorer James Dobson, who beat two men and bent a shot into the far corner from a narrow angle. Two minutes later, Dobson was at it again, trying his luck from 25 yards, but this time Lincoln did well to tip it over.
The Rebels were rampant by this point and threatened to put the game out of sight, but Bognor were awarded a debateable penalty after the referee spotted an infringement by Dan Hicks. Ollie Pearce converted the spot kick on 32 minutes, sending Mark Scott the wrong way.
Bognor had more of the ball after pulling the goal back, and could well have levelled had a white-shirted player connected with a super cross from Pearce.
But the visitors' momentum was halted on the stroke of half time by a double blow. First, midfielder Darren Budd was dismissed for kicking out at Lee Togwell after the two players had challenged for the ball in the centre circle.
Minutes later, on the stroke of half time, Adebayo scored the goal of the game – and a goal of the season contender – as he charged into the box and volleyed Sean Fraser’s cross into the top corner from eight yards.
Slough went in search of a fourth goal in the second half and enjoyed large spells of pressure with their man advantage.
Chances came at regular intervals for the Rebels to extend their lead. Winger Dobson had something of a personal battle with Lincoln as he forced the goalkeeper into a number of good saves, and Charlie Moone was unlucky not to get his name on the scoresheet with several efforts, one of which was ruled out for offside.
Bognor remained a threat, particularly through Pearce. Scott tipped away a free kick from the striker early in the half, and he went close late on when he cut inside and fired just wide of the far post.
But Slough will wonder how they didn't extend their lead, as Dobson hit three more thunderbolts - two of which Lincoln punched clear and the other hit the near post and rebounded to safety – whilst Moone put a volley over the bar.
Bognor stuck to their task well with ten men. James Fraser fired over and Slough had a let-off with six minutes remaining as a shot hit the bar and was scrambled away.
But man of the match Adebayo was not to be denied a debut hat-trick as Slough completed a convincing win. Dobson nodded on for Togwell, who played the ball across goal where the striker swept the ball past Lincoln from eight yards with almost the last kick of the game.
Bognor Regis Town Line up
Dan Lincoln, James Crane (c), Tom Cadman (James Muitt 72), Doug Tuck, Sami El-Abd, Chad Field, James Fraser, Darren Budd, Thomas Byrne (Will Robinson 80), Ollie Pearce, Lorenzo Dolcetti (Ed Sanders 67). Not used: James Wild.