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The Blues fought back from a dismal start and, after Ben Wodskou's opener for the hosts, Feeney fired them level before sub Ajiboye volleyed the winner.
It meant a fourth win in five unbeaten games for Rob Elliot's side, who climbed to 11th in the National League in front of a vocal away following at the ARMCO Arena.
Things did not seem on course for a positive afternoon for long spells as a dire first half ended with Wodskou firing the second-bottom home side in front.
But captain Feeney volleyed Carlisle back on terms in the 69th minute before Ajiboye converted a Callum Whelan cross as Elliot's substitutions took effect.
The win, though, came at a cost with defender Tom Anderson going off injured in the first half.
Morgan Feeney celebrates bringing Carlisle level in the second half (Image: Richard Parkes)
Carlisle, though, are now in their best run of form under Elliot even if their performance in the West Midlands left much to be desired before their improved closing spell.
United were backed by 585 fans amid miserable weather conditions, as Elliot made three changes to the side that started against Tamworth in last week's FA Cup win, handing recalls to Adam Desbois, Feeney and Josh Williams.
They replaced Archie Mair, Kenton Richardson and Stephen Wearne, while Whelan, Micah Obiero and Anderson kept their places in the XI.
There was no place in the squad for Jack Ellis - while Luke Armstrong was with the squad and took part in the warm-up as he nears a return to action.
Luke Armstrong, right, was with the squad at Solihull and took part in the warm-up (Image: Richard Parkes)
Solihull boss Chris Millington made two changes, with new keeper Sam Johnson straight into the side and Ryan Woods also handed a start, with Nat Ford and Joe Sbarra dropping to the bench.
The game began in wet and windy conditions as Carlisle forced the first opening, Regan Linney seeing a shot blocked after Obiero and a home defender went down as they battled for the ball.
Callum Whelan goes close in the first half (Image: Richard Parkes)
At the other end, an inswinging Marvin Johnson corner tested Desbois, who had to punch clear on the stretch, before United gave up possession near their box and Wodskou's shot forced an important block from Anderson.
Carlisle, amid a spell of pressure in the home half, forced a chance for Teddy Jenks, who was unable to hit the target when picked out in the box by Leo Duru's cross.
Johnson tested Desbois on the break as things continued without any sign of an early opener, United forcing several corners to no avail.
There was an opening for the Blues on the break in the 25th minute as Jenks broke past the home midfield and fed Obiero to his left - the striker rounded the advancing Johnson and tried to slot home from a tight angle, but a retreating defender got in the way.
Carlisle were forced into a change soon afterwards as Anderson went down holding his right leg, the defender unable to continue as Richardson took his place.
Tom Anderson went off injured midway through the first half (Image: Richard Parkes)
Things from there remained of poor quality, with long throws from both sides delivering nothing and precious little in the way of quality from both sides as the deadlock remained.
Near the end of the half, Feeney blocked a Johnson shot before Carlisle broke and Linney tested Johnson from the left, Whelan then skimming the woodwork as the hosts cleared.
Yet a sterile half then suddenly ended with the hosts in front.
First, Feeney fouled Jacob Wakeling, and Johnson's free-kick was cleared across the box by Harvey Bunker - Wodskou arriving to fire it across goal and into the corner of United's net.
It was the last meaningful touch of a dismal first half and left Carlisle with some serious work to do.
Early in the second half, Linney slipped as he tried to shoot from the left, while Duru couldn't bring a deep cross under control in the Moors box.
Jenks was next to see a shot blocked as Carlisle tried to attack with greater urgency, Obiero then forcing a good save from Johnson as he shot on the turn.
Micah Obiero rounds the keeper but is unable to convert (Image: Richard Parkes)
Elliot made a triple change on the hour mark as he introduced Ajiboye, Azeez and Ryan Galvin.
And eventually United got themselves level as Feeney struck.
First, Richardson made a determined challenge to prevent the hosts from counter-attacking and, after Azeez's shot was half-blocked, it spun into Feeney's path - and the captain drove a well-taken volley low past Johnson.
Further changes followed, including Wearne for Linney, and Carlisle's substitutions continued to take effect as the Blues kept the pressure on.
And eventually they got in front, as they worked the ball to the right in the 78th minute and Whelan crossed - finding Ajiboye in front of his man and the sub's volley too powerful for Johnson.
The home side tried to respond, Tyler French going close following a scramble from a deep corner.
Obiero was then denied a third for Carlisle by French's block just in front of the line, and Soihull were unable to force a late change as they were consigned to a seventh home defeat in eight games this season - United climbing into the top half of the table.
Solihull: Johnson, Mills, Whitmore, French, Tiensia, Woods (Sbarra 72), Andrews, Johnson (Rutherford 67), High (Hoti 81), Wodskou (Kachunga 81), Wakeling. Not used: Ford, Carson, Pinnington.
Goal: Wodskou 45.
Booked: Wakeling.
United: Desbois, Feeney, Nicholson (Galvin 61), Anderson (Richardson 28), Duru, Williams (Ajiboye 61), Bunker, Whelan, Jenks (Azeez 61), Linney (Wearne 71), Obiero. Not used: Mair, Mullarkey.
Goals: Feeney 69, Ajiboye 78.
Booked: Wearne.
Ref: Kavan Hurn.
Crowd: 1,206 (585 United fans).