Northern Ireland's 3-1 defeat to Netherlands on Friday (AEST) meant Germany could put a three-point cushion between itself and Michael O'Neill's men with a win and they just about did enough to prevail in spite of an awkward start.
Any chance of Germany repeating their 8-0 demolition of Estonia vanished with Can's early red card, as the Juventus midfielder – who was deployed in defence – paid the price for complacency at the back.
But fortune smiled on Die Mannschaft after half-time, as Gundogan found the net twice with deflected attempts and Werner completed the scoring to leave Joachim Low's men level on 15 points with Group C leader Netherlands.
Can lasted just 14 minutes before his moment of madness, allowing a pass across the edge of the box to roll past him and stretching to reach it, fouling Frank Liivak and earning a straight red.
The visitor remained the dominant force, however, with Luca Waldschmidt drawing a good save from Sergei Lepmets in the 24th minute.
Lepmets could only watch and hope just before the break, as Marco Reus' 30-yard free-kick rattled against his crossbar.
But Germany did break the deadlock early in the second half, with Gundogan's low drive from just outside the box deflecting in off Reus.
The Manchester City midfielder struck again six minutes later, this time receiving Reus' cut-back inside the area and seeing his strike flick off a defender and beat the helpless Lepmets.
Substitute Werner's clincher was rather more impressive, as the forward raced on to Gundogan's pass and smashed home after cutting inside to dodge Lepmets.