Two tries from David Strettle, one from Nick De Jager and five points from Spencer's boot had put Saracens into a commanding second-half lead.
But Max Bodilly touched down twice, Gareth Steenson converting both, to peg Sarries back only to lose it cruelly at the death.
And Itoje praised the leadership group at the club who saw them through to clinch the crown.
"I am so happy for everyone involved, it was such a massive team effort. To win it that way was amazing," said 20-year-old Itoje.
"We knew what our plan was and knew they would have their moments but we stuck to it and it has paid off.
"There has been a lot of work that has gone on behind the scenes and the coaches have believed in the team since day one so I'm really thankful we have been able to repay their faith.
"This team has some incredible leaders who make my job easy. They are able to rally the troops and get everyone on board so it was a collective team effort from all those guys."
Meanwhile man-of-the-match Spencer admitted he was more than a little nervous before the game-deciding kick.
"My legs were shaking a bit but I am so glad it went over," he said.
"It does not get better than that. Credit to Exeter, the last 20 minutes they made us work but it just shows what a tight-knit group we have here - I am so proud of the boys."
Ceri Sweeney hit two penalties and Spencer one in a tense first half. Saracens then stormed out of the blocks after the break to take a 14-point lead.
Exeter never gave in, despite facing a mammoth task, and could have stunned Saracens late on to defend their crown.
But Spencer's kick crushed their dreams and despite this Chiefs' skills coach Ricky Pellow was delighted with the fight his young side showed.
"I am massively proud of the group today for the effort they have put in and how we clawed it back - we are disappointed but full credit to Saracens, they put us under a lot of pressure," he said.
"We are always going to fight right until the end and for this group of players they have learnt a lot during this LV= Cup. It has been a great competition for us.
"Hopefully all the things they have learnt will help them when we play Leicester in the league next - that's what is really good about the LV= Cup, you get an opportunity to blood the young guys so they can develop their games."
Saracens: Ransom (Fercu, 27); Ellery, Bosch, Streather (Goode, 19), Strettle; Mordt, Spencer; Barrington (Lamositele, 72), Saunders (Spurling, 62), Figallo (Alo, 53), Itoje, Smith (Kruis, 50), de Jager, Hankin, Joubert (Brown, 50)
Replacements not used: Whiteley
Exeter Chiefs: Bodilly; Vainikolo, Hughes (McGuigan, 41), Naqelevuki (Steenson, 55), James; Sweeney, Thomas (Lewis, 33); Sturgess (Rimmer, 50), Taione (Bateman, 50), Low (Brown, 50), Lees, Sexton, White (Johnson, 59), Conlon (Holmes, 67), Horstmann
Scorers
Saracens
Tries: Strettle 2, De Jager
Conversions: Spencer
Penalties: Spencer 2
Exeter Chiefs
Tries: Bodilly 2
Conversions: Steenson 2
Penalties: Sweeney 2
Referee: Ian Davies
Attendance: 8,365
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