The centre-back - on loan from Rangers - found the net in the 72nd minute to cancel out Merouane Zemmama's first-half opener.
United finished the match with 10 men after Prince Buaben was shown a straight red card by referee Mike Tumilty following an off-the-ball incident involving Hibernian striker Anthony Stokes.
Hibs made a lively start as they put their visitors on the back foot inside the first minute. Stokes found Derek Riordan in the box with a nice return pass but the striker's delivery across the face of goal eluded all his team-mates and outstretched goalkeeper Nicky Weaver.
The threat remained, however, and Weaver had to protect his near post as Zemmama's shot from a tight angle hit of the outer side of the woodwork.
After Paul Dixon failed to find the target with a free-kick 25-yards from goal, United registered the next attempt on goal in the ninth minute.
Webster rose high to meet Craig Conway's corner but his header sailed over the bar.
A long kick up the park from Hibs goalkeeper Graham Stack fell to Stokes inside the box after the United defence failed to deal with the high ball.
The former Sunderland player's shot was expertly blocked by Weaver and his attempt on the rebound came back off the side of the post before going behind.
Hibs took the lead in the 27th minute after United were made to pay for slack defending.
A loose pass back to Darren Dods was intercepted by Zemmama and the Moroccan international showed great composure to turn Webster in the box before drilling a left-footed shot off the underside of the bar and into the net from close range.
Hibs were keen to take advantage of a fragile-looking United defence as Weaver was called into action to block David Wotherspoon's low drive.
Dominant Hibs showed no signs of letting up in their pursuit of a two-goal cushion with Zemmama giving United left-back Dixon a torrid time with his marauding runs.
But United looked a different side at the beginning of the second period after their half-time team talk from manager Craig Levein.
Craig Conway's corner was headed back into the cluster of players by Dods but Mihael Kovacevic's overhead kick was off-target.
Buaben was then allowed to run at the Hibs rearguard before unleashing a stinging shot that drifted just wide.
It was one-way traffic as Hibs were unable to recapture the kind of from that saw them outplay United in the opening 45 minutes.
Morgaro Gomis should have done better with a left-footed shot inside the box that was easily saved by Stack.
United made a change with David Goodwillie replacing Danny Cadamarteri. The Scotland Under-21 striker was involved straight away as he forced a save inside the box from Stack after showing neat close ball control to make room for a shot.
United were forced into a change in the 64th minute when Dixon had to be taken off on a stretcher after sliding in to cut out Wotherspoon's attempted cross.
Just moments after his arrival, Sean Dillon was beaten by Zemmama and the winger crossed for Riordan but the striker was guilty of a glaring miss as he volleyed over from six yards.
United restored parity in the 72nd minute as Webster notched his second goal in as many games with the aid of a deflection.
His shot from 20 yards caught Stack off balance before finding the net via the inside of the post.
United were relieved not to be behind again after Zemmama's free-kick from the corner of the area flashed across goal.
The visitors then had claims for a penalty waved away by referee Tumilty after Conway fell inside the area as he attempted to make contact with Gomis' cross.
A hard-fought match was marred as Buaben was shown a straight red card two minutes from the end following an off-the-ball incident also involving Stokes that was witnessed by an assistant referee.