Masson netted on the half-hour, applying the finishing touch to a cross from Josh Magennis, with the latter also hitting the post before the break.
Shiels then equalised eight minutes into the second half, scoring with a diagonal finish following a pacy counter-attack and pass from William Gros.
Aberdeen piled on the pressure in the closing moments and forced home goalkeeper Cammy Bell into an important save from Ryan Jack, but the teams had to settle for a share of the points.
Kilmarnock and Aberdeen each made three changes for Saturday's game - home manager Kenny Shiels handed starts to Gros, Lee Johnston and Mohamadou Sissoko in place of Dieter van Tornhout, Danny Racchi and Ryan O'Leary, all of whom were on the bench.
Jason Brown returned in goals for Aberdeen after three matches, with Jamie Langfield dropping to the bench; Rory McArdle and Rory Fallon also started in place of Russell Anderson and Fraser Fyvie as both sides sought to return to winning ways after midweek home defeats.
The hosts were the livelier of the two teams in the opening stages, Gros trying to carve a chance inside the first minute, and Dean Shiels tried a shot from outside the right edge of the box which went harmlessly over on nine minutes.
Kari Arnason responded with an even more speculative effort moments later, but the Aberdeen midfielder was closer to the target with a reaction shot from Magennis' pass from the right corner. In between, Lewis Toshney was needed to head clear Ryan Jack's inviting cross into the box.
Fallon sent the ball well over the goal midway through the first half after Shiels gave away possession with a misplaced pass - then Kilmarnock had the best chance yet of the opening goal when Ben Gordon sprinted up the left wing and squared for James Fowler, but he shot straight to Brown.
Arnason struck a free kick on 25 minutes, but his effort cannoned off the defensive wall positioned a yard inside the box, and Andrew Considine was then flagged offside as he chased a rebound ball played out to the right.
Aberdeen took the lead on the half-hour - Jack and Magennis linked on the right for the latter to cross into the six-yard box, and although the chance looked as though it might be lost when Fallon and Scott Vernon both failed to turn it home, the ball fell for Masson to score at the left-hand post.
Shiels played a good one-two to get himself into a scoring position but Brown safely clutched his effort. At the other end, Magennis tried a shot after some pacy Aberdeen passing and saw his shot hit the base of the right-hand post.
Jack delivered a corner from the right which saw Fallon head into the arms of home goalkeeper Bell.
Liam Kelly tried his luck for Kilmarnock with a shot inside the first minute of the second half which Brown clutched at the top-left corner.
The hosts made all the early running and were rewarded with an equaliser on 53 minutes when Gros supplied Shiels with a precise pass in from the left wing, and the forward bore down on goal before striking and scoring with a diagonal finish which went in at the right-hand post.
Aberdeen tried to re-establish their lead, with Fallon heading over from a cross after a lengthy run by Considine, and Jack passed inside the box to Vernon, but he was unable to connect properly.
Vernon then had the ball in the net just after the hour but had already been flagged offside, and Fallon rose well but headed into Bell's arms from a corner.
Gros unleashed a shot from the left edge of the box which surprised the Aberdeen defence and was blocked by a grateful Brown, but although the visitors never managed to clear their lines, the was eventually blasted over by Danny Racchi.
At the other end as the game entered its final 10 minutes, Vernon passed along the edge of the box for Arnason, who saw his shot saved by Bell.
Jack lofted in a shot from outside the box three minutes from time which Bell had to jump to beat away from the top left-hand corner, and substitute Dieter van Tornhout was off-target with a low drive in injury time.