Whelan scored his second goal of the competition in the first half
Stoke moved into the quarter-finals of the Carling Cup with a comfortable win over League Two outfit Rotherham.
The home side took the lead on 19 minutes when midfielder Glenn Whelan lifted a free-kick over the wall from the edge of the box.
And Danny Pugh doubled the lead on 59 minutes when he rifled a shot past goalkeeper Andy Warrington.
Dave Kitson had a goal disallowed when he was adjudged offside after Whelan's shot was saved 11 minutes from time.
Stoke's victory ended Rotherham's hopes of a high-profile scalp after knocking out Championship sides Sheffield Wednesday, Wolves and Southampton in previous rounds.
Potters boss Tony Pulis made nine changes to the side that drew 0-0 with Wigan on Saturday, with only Ryan Shawcross and Seyi Olofinjana retaining their places in the starting XI.
And predictably the Premier League side dominated the early stages of the match, with Whelan, Shawcross and Pericard guilty of missing good opportunities to open the scoring.
But Whelan made amends for his earlier miss when Alex Rhodes was penalised for a high foot on Pugh, sending his 20-yard free-kick past Warrington.
Boosted by their 2,000-strong travelling fans, Rotherham responded when Nick Fenton's far-post header just cleared Steve Simonsen's bar.
Simonsen kept his side in front after the interval when he prevented Reuben Reid's fine curling shot from sneaking past his far post.
But the Potters doubled their lead when Kitson's pass put Pugh through.
Kitson thought he had added a third on 79 minutes when Warrington spilled Whelan's drive, but the assistant referee's flag meant his celebrations were cut short.
Stoke boss Tony Pulis:
"Whoever we get now in the last eight we will handle it. The big sides are still there, but we'll not be frightened of that.
"I just want another home draw, we will enjoy that. It is the first time for a very long time that this club has been in the finals of a major competition.
"We are 12th in the league and in the league cup quarter-finals, it is absolutely fantastic times for us."
Rotherham boss Mark Robins:
"We wanted to go as far as we could, but it was a bridge too far. Everyone connected with the club should be proud of what the players have achieved. And we are still in two other cup competitions.
"It is important that we have good cup runs, it is something we spoke about right at the outset when we had our 17-point deduction.
"That could have killed our season, it has been very difficult to recover from that and we did it very quickly. It has been very tough mentally to come back from that deficit. The cup ties have been a welcome relief."
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