Flanker Ross Moriarty got England's second try with half an hour on the clock and the visitors went in 13-0 up at the break.
A new-look Ireland came out after the break and the hosts soon shot into a 15-13 lead courtesy of two tries within 10 minutes and a well-executed drop goal from number ten Conor McKeon.
But England regained composure and Northampton Saints man Olver slotted a drop goal of his own to put England 16-15 ahead.
There were a couple of nervy moments for the men in white as Ireland saw two further drop goal attempts sail wide as England held out for the win.
Head Coach Fletcher said: "To be honest we should have been out of sight by half time, we scored two tries but it should have been more - we left a few out there.
"In the second half Ireland came back fighting and we seemed to lose some of our composure, two guys went into the sin-bin [Gus Jones and Danny Hobbs-Awoyemi] and we were staring a defeat in the face at 15-13 down.
"Full credit to guys for battling back and putting Sam in a position to get his drop goal. Overall it's been a good experience, a good exercise, and a good test for our players."
Points: Burns 1T, Moriarty 1T; Olver 1P, 1DG
England Under 19 starting XV against Ireland
15 Nathan Earle (Saracens)
14 Zach Kibirige (Newcastle Falcons)
13 Tom Stephenson (Northampton Saints)
12 Javiah Pohe (Leicester Tigers)
11 Josh Watkins (Worcester Warriors)
10 Sam Olver (Northampton Saints)
9 Jordan Burns (Harlequins)
1 Danny Hobbs -Awoyemi (Northampton Saints)
2 Harry Thacker (Leicester Tigers)
3 Adam Jamieson (Bath Rugby)
4 Maro Itoje (Saracens)
5 Tom Ellis (Bath Rugby)
6 Ross Moriarty (Gloucester Rugby)
7 Gus Jones (London Wasps)
8 Hayden Stringer (Saracens)
Replacements: (all used)
16 Cameron Neild (Sale Sharks)
17 Alex Lundberg (London Wasps)
18 Jake Armstrong (Leeds Carnegie)
19 David Hughes (London Wasps)
20 Lewis Ludlow (Gloucester Rugby)
21 George Tresidder (Leicester Tigers)
22 David Buck (Bath Rugby)
23 Tom Hudson (Leicester Tigers)