Stuart Armstrong had put the Tannadice men ahead after 29 minutes as it appeared Fenlon's 18-month spell as manager was in real jeopardy.
The Leith side have suffered the humiliation of a 9-0 aggregate defeat to Europa League rivals Malmo and defeat in last week's Edinburgh derby already this season, and the reaction to the Terrors' opener was a chorus of furious boos from the home support.
One fan in the Main Stand even had to be restrained after darting from his seat to berate chairman Rod Petrie in the directors' box.
But Robertson's stunning leveller may yet prove to be Irishman Fenlon's salvation as Hibs finally collected their first Scottish Premiership point of the campaign.
If Fenlon hoped for a swashbuckling display to win over his detractors, he was let down by a messy opening 20 minutes in which neither side looked capable of stringing more than a pass or two together.
It was a spell where even the most basic control looked a task too far for some of the players and where bodies inevitably ended up tangled together as moves broke down.
As such, Hibs right-back Fraser Mullen collected the game's first caution for a wild lunge on Gary Mackay-Steven.
Ryan Dow did split the home defence after a quarter of an hour but David Goodwillie's first touch stuck under his feet and Michael Nelson raced in to cover.
The Scotland striker's control again let him down as Mackay-Steven found him at the back post but he recovered to wriggle free on the bye-line before cutting back for Armstrong, whose shot was blocked by Robertson's last-ditch block 10 yards from goal.
John Rankin then fired over from Mackay-Steven's cut-back after he skipped his way past the befuddled Mullen but United's pressure soon told as Armstrong put them ahead just before the half-hour mark.
Ryan Butcher sent a 40-yard crossfield pass to the Scotland Under-21 skipper on the edge of the Hibs box and with the Leith back four nowhere to be seen, he was able to make up for a heavy touch by sliding the ball beyond keeper Ben Williams into the corner of the net.
The angry boos from the frustrated home support were to be expected but Fenlon tried to take the initiative by replacing the toiling Mullen with Ryan McGivern six minutes before the break.
James Collins' 25-yard strike, beaten away by Radoslaw Cierzniak, was the best the home side managed before the break and Fenlon was again treated to the sound of his own supporters telling him exactly what they thought as the whistle went for half-time.
The chances continued to fall for United, with Goodwillie wasting Sean Dillon's centre with a miscued header after 56 minutes while Andrew Robertson found the side-netting after racing forward from left-back.
But eight minutes later the game exploded as Thomson and Gunning were shown straight red cards.
A bouncing ball on the edge of the United box saw the pair come together before Thomson reacted furiously to the Arab defender's robust challenge.
But even after referee Crawford Allan cleared a scrum of players to flash red at both, the pair were at it again as they were separated once again on their way off the pitch.
The confrontation seemed to raise Hibs' hackles and Liam Craig took his frustrations out on the ball by lashing a fierce drive at Cierzniak, who had to push it over his crossbar.
Williams then denied United substitute Brian Graham after he was sent clear by Ryan Gauld but the save proved crucial as Hibs levelled with nine minutes left.
Robertson first tried to hit one from 25 yards out but saw it blocked by a United leg. The former Scotland midfielder remained alert, however, and smash the rebound into the bottom corner.
- Sports & Recreation
- Kevin Thomson