After asking India to follow on, England managed three second-innings wickets yesterday as the tourists closed on 129 for three - still trailing by 162 runs.
Swann claimed two of the wickets to fall, including first-innings centurion Rahul Dravid, as his slow right-arm spin looked set to be crucial to England's hopes on a wearing final-day wicket.
After a summer when England's seamers have done most of the damage on conducive wickets, the spin bowler acknowledged he was looking forward to finally getting his chance to play a key role.
"It excites me - especially after you play a whole season on belting batting and seaming wickets," he told Sky Sports. "It is nice to get a spinning wicket. We need to make inroads in the morning. We've got 20 overs where the ball will talk a lot. It was easier today once we got the hard ball. It died a bit early on.
"I was fairly happy with the way I bowled all day. My body is aching like it hasn't all summer, but it's a nice ache."
Dravid was disappointed to fall for 13 before the close last night after he batted through India's first innings to be unbeaten on 146.
Swann eked him out yesterday evening with a ball Dravid acknowledged he had just nicked before being caught by Alastair Cook at short leg.
India's batting has relied upon his defiance in the series, but Dravid believes star man Sachin Tendulkar - who reached stumps unbeaten on 35 - can finally find form today and save the tourists from a whitewash.
"It's been a tough series," he said.
"They are a quality side and have bowled well as a group. Some of us have had a tough series, but hopefully tomorrow it is Sachin's day.
"If the other guys can rally around him maybe we can save this Test."
England were pushed into the unfamiliar territory of taking the second new ball in the India first innings, almost single-handedly by Dravid but also by the defiance of Amit Mishra and RP Singh.
The shame for India was that only one frontline batsman lived up to his billing on a perfectly feasible pitch.
Pushed up to open for the second time in the series, Dravid reattuned himself to the position expertly in the absence of Gautam Gambhir - who was able to bat eventually at number nine, having suffered concussion on Friday.
MS Dhoni kept Dravid company for only 12 overs yesterday morning, his fidgety innings ending when James Anderson struck with his 10th ball - a routine caught-behind, off an outswinger.
Dravid's only moment of concern came when a direct hit from Kevin Pietersen would have run him out on 61, after a mix-up with Dhoni over an aborted single.
He plundered the last three of the 15 boundaries in his 168-ball hundred in one Swann over, then reached the milestone - for the 35th time in Tests - with a dab to third-man for two off Tim Bresnan.
Dravid has made one more Test century than his childhood hero and India great Sunil Gavaskar, is fourth on the all-time list, only the third from his country to bat throughout an innings at this level and the only member of this touring party to have reached three figures this summer.
That equates to seven England hundreds, one of many jarring statistics which are symptomatic of an unexpectedly one-sided Test series.
Mishra batted admirably in a stand of 87 either side of lunch, headlining his innings by smashing Swann over long-on for six from the last ball of the morning.
It took a memorable piece of cricket from England to shift him, deploying Ian Bell at short square-leg where he took an outstanding one-handed catch away to his right from a pull at Bresnan (three for 54).
Gambhir dealt poorly with a short ball into his body from Stuart Broad, which resulted in an easy catch looped to gully.
But Singh then dominated a ninth-wicket stand of 36 to at least carry India to 300 for the first time in the series - before Bresnan returned to have him well-caught at third slip and then last man Shantha Sreesanth at cover in the space of three balls.
As India started again six overs before tea, it was not a great surprise to see Dravid resuming in Gambhir's published place alongside Virender Sehwag at the top of the order.