Paul Jewell made 11 changes to the Wigan team that drew 1-1 with Middlesbrough on Sunday and they were far from convincing. But the crucial goal finally came with just four minutes remaining, when Roberts raced on to a flick-on from fellow substitute David Connolly and then tucked the ball past goalkeeper Gareth Stewart.
The Cherries were without 11 players through injury and Josh Gowling through suspension, leaving manager Sean O'Driscoll with just four substitutes, but the visitors started brightly.
Bournemouth saw a Brian Stock shot fly off target and it took 12 minutes for Wigan to threaten, but striker Henri Camara failed to get a shot away from close range after he had collected Gary Teale's cross.
Josip Skoko, who joined from Turkish side Genclerbirligi just before the transfer deadline, made his first contribution of note midway through the opening period with a powerful run from midfield, but his square pass to Alan Mahon was cleared easily.
A Mahon corner was then glanced wide by Andreas Johansson from the edge of the six-yard area and Camara fired straight at Bournemouth goalkeeper Stewart as the hosts began to shake off their slumbers.
Bournemouth went agonisingly close to breaking the deadlock six minutes after the break but James Coutts flicked a header wide of goal from Shaun Cooper's neat cross.
Wigan captain Matt Jackson had stayed forward for a corner and headed wide from Ryan Taylor's chip to the far post before Andrew Surman hit a long-range effort straight at John Filan at the other end.
Jewell threw on strikers Roberts and Connolly for Johansson and Teale and almost immediately, Roberts robbed Neil Young and laid the ball off to Camara, whose cross picked out Skoko at the far post, only for the midfielder to head straight at Stewart.
In the 68th minute, Roberts missed a golden chance when Cooper inexplicably passed the ball to him, but the former West Brom forward clipped his shot wide after skipping past the defender.
Camara was next to miss, nodding wide from Mahon's drilled centre, as the frustration continued, before Mahon gave Stewart a couple of routine saves.
Wigan had a scare with 14 minutes of normal time remaining when David Wright headed Steven Foley's in-swinging cross against his own post. Stewart then pulled off a superb low stop from Camara's close-range shot following Connolly's pull-back before Roberts finally scored in the 86th minute. James Hayter missed a last-gasp chance for Bournemouth but Wigan clung on.
Wigan: Filan, Wright, Jackson, Thome, McMillan, Teale, Skoko, Taylor, Mahon, Johansson, Camara. Substitutes: Walsh, Kavanagh, Connolly, Roberts, Henchoz.
Bournemouth: Stewart, Cooper, Young, Hart, Coutts, Surman, Stock, Foley, Hayter, Keene, Rodrigues. Substitutes: Pitman, Whisken, Cleverley, Hudson.
Referee: N Miller (Durham).
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