A method of making a vacuum insulating glass unit, the method comprising: having first and second glass substrates (60a, 60b), the first substrate including a hole therein, the hole having first and second portions (54a, 52), the first portion being closer to an outer surface of the first substrate than the second portion, the first portion having a first width across the first substrate and the second portion having a second width across the first substrate, the first width being wider than the second width, the first and second portions (54a, 52) together forming a through-hole through the first substrate; placing in the hole a flanged pump-out tube, the flanged pump-out tube including a tube having an inner diameter and an outer diameter, with the walls of the tube being defined therebetween, the outer diameter being narrower than the second width, the flanged pump-out tube including a skirt, the skirt having a diameter, the diameter being wider than the second width and being narrower than the first width, wherein the skirt is intermediate upper and lower ends of the tube so that the skirt is spaced a first distance from an upper end of the tube and spaced a second distance from a lower end of the tube with a length, wherein placing the skirt extends the lower portion of the tube at least partially into the second portion of the hole and locates the upper portion of the tube at least partially in the first portion of the hole, wherein the skirt sits in the first portion of the hole, and wherein the tube is sealed in the hole using a frit; sealing an area proximate a periphery of the first and second substrates (60a, 60b) with an edge seal provided around peripheral edges of the first and/or second substrates (60a, 60b), a cavity being defined between at least the first and second substrates (60a, 60b), and a plurality of spacers being provided between the first and second substrates (60a, 60b) in the cavity and helping to maintain the first and second substrates (60a, 60b) in substantially parallel, spaced-apart relation to one another; evacuating the cavity to a pressure less than atmospheric pressure; and heating the pump-out tube so as to cause at least a part of the upper portion of the tube to collapse inwardly upon itself, covering the second width and hermetically sealing the VIG unit and forming a sealed tube, the sealed tube being flush with, or partially recessed with respect to, an outermost surface of the first substrate. having first and second glass substrates (60a, 60b), the first substrate including a hole therein, the hole having first and second portions (54a, 52), the first portion being closer to an outer surface of the first substrate than the second portion, the first portion having a first width across the first substrate and the second portion having a second width across the first substrate, the first width being wider than the second width, the first and second portions (54a, 52) together forming a through-hole through the first substrate; placing in the hole a flanged pump-out tube, the flanged pump-out tube including a tube having an inner diameter and an outer diameter, with the walls of the tube being defined therebetween, the outer diameter being narrower than the second width, the flanged pump-out tube including a skirt, the skirt having a diameter, the diameter being wider than the second width and being narrower than the first width, wherein the skirt is intermediate upper and lower ends of the tube so that the skirt is spaced a first distance from an upper end of the tube and spaced a second distance from a lower end of the tube with a length, wherein placing the skirt extends the lower portion of the tube at least partially into the second portion of the hole and locates the upper portion of the tube at least partially in the first portion of the hole, wherein the skirt sits in the first portion of the hole, and wherein the tube is sealed in the hole using a frit; sealing an area proximate a periphery of the first and second substrates (60a, 60b) with an edge seal provided around peripheral edges of the first and/or second substrates (60a, 60b), a cavity being defined between at least the first and second substrates (60a, 60b), and a plurality of spacers being provided between the first and second substrates (60a, 60b) in the cavity and helping to maintain the first and second substrates (60a, 60b) in substantially parallel, spaced-apart relation to one another; evacuating the cavity to a pressure less than atmospheric pressure; and heating the pump-out tube so as to cause at least a part of the upper portion of the tube to collapse inwardly upon itself, covering the second width and hermetically sealing the VIG unit and forming a sealed tube, the sealed tube being flush with, or partially recessed with respect to, an outermost surface of the first substrate.