It was the hosts who had the first chance of the game on six minutes when Fonguck’s low cross failed to find a Barnet man and almost rolled into the far corner.
Debutant Manny Duke tried his luck from distance two minutes later after a failed clearance from Ryan Boot fell kindly for him but his effort cannoned over.
The Bees caused even more problems on 12 minutes when Fonguck fizzed in a cross from the left which found Jack Taylor in the box but his shot fell wide of the post.
Soon after, Duke charged forward from the Barnet half before feeding a delightful ball into Josh Walker but Boot was on hand to race out and intercept.
It was Solihull’s chance to pose a threat on 20 minutes when Adi Yussuf’s spectacular acrobatic effort from the edge of the area forced Mark Cousins to tip the ball over.
A lovely back heel flick from Duku found David Tutonda who fed the ball back to the debutant, however, his effort from close range flew narrowly over the bar on the half-hour mark.
After a first half that saw Barnet have the better of the chances, they were first to pose a threat in the second after Tutonda found Fonguck in the box but he couldn’t find the net.
However, it was the Barnet number 18 who broke the dead-lock just shy of the hour mark, burying in Walker’s rebounded effort from close range, giving the hosts a deserving lead.
John Still’s men could have doubled their lead soon after when Cheye Alexander’s cross was cleared to the feet of Duku; his strike from inside the area could only find the arms of Boot.
Solihull substitute Jermaine Hylton tried his luck from distance with just under 20 minutes remaining but his effort whistled wide of the post.
Duku picked up the first and last booking of the game in stoppage time after taking out a Solihull man on the right flank.
Barnet made sure of the points deep into added time when Jack Barham intercepted a Solihull free-kick before playing in Fonguck who twisted and turned before slotting home his second.
The Bees return to league action on 27th October when they travel to Barrow as they aim to clinch their second win on the bounce.
Barnet XI: Cousins, Alexander, Robson (C), Adams, J.Taylor, Fonguck, Walker (Barham), Tutonda, Sweeney, Jules, Duku (Mason-Clark).
Unused Subs: Matrevics, Sparkes, Elito.
Attendance: = 1054 (116 visitors)