Boro had to come from behind after a defensive error left Liam Davis racing free to score the opener for Yeovil after just four minutes.
Jacob Butterfield extended Boro's lead after the break before Kei Kamara's header put them out of sight.
Yeovil stunned the home side when they took their fourth-minute lead.
Home skipper Rhys Williams failed to cut out a long header forward leaving Davis free to race forward and shoot through the legs of the advancing Jason Steele.
Boro were level after eight minutes as a corner was half-cleared and Adomah chipped it back to the edge of the box where Leadbitter sent a half-volley bouncing through the crowd and into the net.
Yeovil were then almost behind on 12 minutes when Boro twice rattled the woodwork in quick succession.
Adomah crossed from the right for Kamara to first thump a header down off the crossbar and then stab the rebound against the post before it was scrambled away.
Yeovil threatened on 19 minutes as Davis robbed Kamara on the right and sent Paddy Madden past Ben Gibson and into the box but the defender recovered and tackled before the striker could shoot.
Boro went ahead from the penalty spot on 24 minutes after over-lapping full-back George Friend was brought down by Joel Grant.
Goalkeeper Hennessey did well to push Leadbitter's spot-kick onto the post but Adomah reacted first to stab the rebound home.
Yeovil had a chance on 34 minutes as a corner landed at the feet of defender Byron Webster but he lashed it high over the crossbar from 10 yards out.
Then on 36 minutes Yeovil had the ball in the net as Ed Upson broke free and when his shot was saved Madden netted the rebound only for the flag to go up.
Boro started brightly after the break and deservedly extended their lead on 50 minutes.
Butterfield played a neat one-two with Leadbitter to find space 25 yards out then slammed his low shot inside the far post.
Yeovil almost pulled one straight back two minutes later as Davis cut down the right and squared for Sam Hoskins to skip into the box and smash in a 12-yard shot that beat the goalkeeper but came back off the post.
Boro then enjoyed a long spell of patient possession before scoring a fourth on 74 minutes, Kamara powering home a header from a Mustapha Carayol corner.
It should have been 5-1 on 81 minutes as Carayol sent Friend surging forward into the box to square for Kamara but somehow he scooped over from eight yards out.
Yeovil missed a sitter on 86 minutes as substitute Kevin Dawson fired in a low ball from wide on the right and Kieffer Moore - two yards out and only needing to poke it in - somehow managed to hook the ball clear.