Farrell nudged Saracens 9-6 ahead in the first half, Burns missing a drop-goal attempt just before the interval, and then stretched his side's advantage to six points early in the second.
But Burns levelled matters - and repeated the dose three minutes from time after Sarries had again edged ahead in the 67th minute.
Saracens were forced to make an 11th-hour change as Farrell moving up off the bench to replace Charlie Hodgson, who became a father for the third time during the early hours with the birth of son Jack.
Gloucester restored Mike Tindall to their midfield after resting him for their home win over Wasps.
England interim head coach Stuart Lancaster was at Vicarage Road to assess the likes of Alex Goode, Brad Barritt and Farrell in the home side and Charlie Sharples, Henry Trinder and Burns from the visitors.
Farrell got the chance to kick-start the scoreboard when referee Sean Davey awarded the hosts a penalty for a ruck offence after six minutes, but he hooked his kick into the driving rain.
Jim Hamilton was penalised at the breakdown three minutes later, and Farrell made no mistake from in front of the posts.
Burns levelled matters on 12 minutes with a difficult kick from wide on the left wing, but Farrell struck again on 18 minutes after Peter Buxton was penalised at a lineout.
Considering the atrocious conditions, both teams tried to play expansive rugby, with sometimes predictable results in terms of handling errors.
One such led to a penalty for Gloucester but although Burns managed the distance from just inside his own half, the ball faded right feet from the posts.
At this stage, however, Gloucester had reversed the earlier trend and were taking the game to Saracens while also getting more of their own way in the scrum, while the home side were justifying their reputation as their meanest defence in the Aviva Premiership.
But the pressure told two minutes from half-time when Davey spotted an offside, and this time Burns hit it straight between the sticks from 47 metres.
Half-time did no favours to Gloucester, who lost some of their impetus upon the restart. Davey then penalised Hamilton for barging and Farrell's fourth goal on 49 minutes edged them further ahead, only for the equally dangerous Burns to nail his third penalty on 53 minutes.
And when another Gloucester scrum forced Saracens into a mistake on 61 minutes, Burns hauled his team level with another long-range shot.
Farrell had the chance to regain the lead four minutes later but his attempt flew across the face of the goal and just wide.
Sharples put his team in peril on 68 minutes, taking the ball into contact deep in his half then not releasing when tackled. Farrell's fifth goal roused the soaked home crowd.
Aside from the duel between Farrell and Burns there was little else to lift Lancaster out of his seat from other contenders.
And when Saracens blundered two minutes from time, Burns was handed a 45-metre opportunity to equalise and delivered in style.
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