Defender Robinson opened the scoring seven minutes after the restart with a neat finish from a corner and then repeated the trick in the 71st minute with a similar effort to boost the Lions' hopes of making the play-offs.
There was still time for Andros Townsend to claim a late third for the visitors, who were good value for their victory after an outstanding second-half display.
Burnley went close in the sixth minute when Alan Dunne fouled on-loan Aston Villa striker Nathan Delfouneso and Chris Eagles lined up a free-kick 30 yards from goal.
The former Manchester United trainee hit a fierce effort which Dunne did well to block, but the hosts began to build up a head of steam.
In the 12th minute Liam Trotter fouled on-loan Chelsea midfielder Jack Cork and Clarets right-back Tyrone Mears flighted a dangerous free-kick into the danger area.
Delfouneso, who scored the winning goal on his debut in the midweek 1-0 victory at Hull, rose highest to connect with Mears' delivery but his firm header came back off the bar from 12 yards as Millwall survived again.
In the 14th minute, Lions striker Steve Morison engineered the space to hit a 25-yard effort which flew narrowly over the bar.
However, the momentum remained largely with the hosts and in the 19th minute Jay Rodriguez hit a 25-yard shot which drew a comfortable save from Millwall goalkeeper David Forde.
Clarets left-back Daniel Fox then went close with a long-range strike but Millwall were largely comfortable as they kept Burnley at bay with some stout defending.
Yet as the first half progressed, Burnley's goal threat increased and Dean Marney went close twice in quick succession with a long-range shot and a header over from six yards.
Morison then drew a routine save from Grant before James Henry and Townsend also went close for the visitors.
Burnley sought to up the tempo after the restart as winger Ross Wallace had a shot blocked by Darren Ward and Marney was denied by Forde.
Yet the Clarets were hit with a sucker-punch in the 52nd when Millwall opened the scoring.
An inswinging corner from Townsend found Robinson six yards from goal and he fired home left-footed.
Millwall almost doubled their lead four minutes later when Henry found Morison but the striker dragged his shot just wide from 12 yards out.
Shortly before the hour mark, Eagles was replaced by Chris Iwelumo as the hosts attempted to turn the screw on the Millwall defence.
However, Robinson claimed his second with 19 minutes remaining when a corner from Townsend was played to Dunne and his delivery from the right flank found Robinson 12 yards from goal.
Robinson kept his composure to strike a left-footed shot into the bottom from 12 yards out.
And Townsend grabbed a third in the 87th minute when he rifled home a low left-foot shot from 12 yards.