Home striker Burchill took advantage of an error by Hibs goalkeeper Yves Ma-Kalambay to put his side in front nine minutes after the break.
However, it was to prove a short-lived lead as Stokes equalised soon afterwards with a good finish. His goal extending the Edinburgh side's unbeaten league run to 11 games.
Hibs fielded an unchanged side following their 2-0 home win over Motherwell, and Rugby Park manager Jim Jefferies made one change to the team which gained the club's first league away win at McDiarmid Park, Danny Invincibile coming into the team in place of Steven Old.
Both sides had their first sight of goal courtesy of early free-kicks. First Garry Hay's curling effort for Killie was headed to safety and then Hibs Merouane Zemmama's 25-yard shot, after he had cheekily disposed Kilmarnock from their own free-kick and set off on a blistering run upfield, flew well over Mark Brown's crossbar.
Kilmarnock were unfortunate not to go in front two minutes later when Mehdi Taouil played the ball wide on the left for Hay to send an excellent cross into the box - Kevin Kyle met it with a solid header but Ma-Kalambay turned it round the right-hand post.
Colin Nish nodded wide from a David Wotherspoon cross, but Ma-Kalambay was again called into action when Invincibile hit a shot towards the right-hand post which the Hibs goalkeeper blocked with his legs.
Burchill then won a corner off Chris Hogg which was swung in from the left by Hay, but Kyle's downward header was put out for another set-piece on the opposite side which this time was immediately cleared to safety.
Hibs enjoyed their share of possession and Zemmama was a constant threat, but a combination of misplaced passes and solid Kilmarnock defending meant they were unable to find a way through.
The best action of a drab second quarter came just before the break - Zemmama's quick cross from a tight angle on the right byline was missed by Nish and Riordan, and then Wotherspoon tried a snap-shot in the last action of the half which Brown dived to smother.
Moroccan Zemmama was again involved early in the second half when he volleyed narrowly over, but not long after Kilmarnock took the lead.
On 54 minutes Burchill took full advantage of a costly error by Ma-Kalambay to bundle home.
Jamie Hamill fired a cross from the right into a busy penalty area and, although the goalkeeper rose as though in command of the situation, he spilled the ball horribly and Burchill was the first to react.
Ma-Kalambay had hearts in mouths again moments later when he missed a Hay free-kick from the left and Frazer Wright was allowed to head narrowly wide of the left-hand post.
In between those efforts in a breathless second half, Bamba strode forward unchallenged from midfield and sent a raking drive just wide.
However, Hibs were back on level terms within five minutes of the hosts' opener when Stokes ran onto a Nish ball down the right and drilled home a low effort into the bottom corner of Brown's net.
Hibs then had David Wotherspoon to thank for booting a Kyle effort off the line.
Nish himself had a penalty shout correctly turned down on 70 minutes after he took exception to a challenge from Fowler and both sides desperately tried to claim a winner.
The hosts came closest to scoring when Taouil's cross was met by a looping Kyle header which Ma-Kalambay just managed to clutch in his outstretched arms under the crossbar.
Craig Bryson earned a shooting chance when Hamill's cross was played back to him in the centre by Kyle, but unfortunately for him the effort came off team-mate Burchill.
Zemmama tried a late half-volley from 15 yards but it cannoned off the blocking Ford and Taouil sent a drive whizzing past the right-hand post.
Derek Riordan had the last word in stoppage time, meeting a clearance on the edge of the penalty box but his volley could only smash the crossbar.