After David Dunn was brought down in the penalty area early into the second half, Kelly stepped up to put Latics one goal ahead.
Collins struck on 70 minutes to level the game but the point means Latics maintain their unbeaten start to the season.
An injury to Joseph Mills in the warm up meant Timothee Dieng got his first start of the season.
Latics started slowly with Shrewsbury Town dominating the opening proceedings.
They had two good chances to take the early lead as Martin Woods’ corners found the head of Jermaine Grandison and Mark Ellis but neither could guide the ball on target.
Athletic started to get a grip of the game after their slow start with experienced midfielder David Dunn dictating the play from the middle of the park.
He slid the ball through to Jonathan Forte five minutes into the game but the lone striker could not take the ball in his stride and the chance was lost.
After Abu Ogogo received the first booking of the game from a late challenge on Liam Kelly, Dunn attempted to dance through the Shrews defenders but he was just stopped in his tracks as he continued his bright start to the game.
The best chance of the first half came on 15 minutes through Forte. The striker received the ball in the box from Kelly, he decided to take the shot early but his volley rebounded off the crossbar as Latics went closer.
Latics were continuing to dominate possession but were struggling to create clear cut chances in attack.
The visitors did have chances in the first half but neither James Collins’ or Woods’ shot troubled goalkeeper Joel Coleman.
With five minutes of the first half remaining, Danny Philliskirk slipped the ball through to the onrushing Forte but out of nowhere, Grandison, who looked solid at the back, blocked.
Both sides had chances before the break to take the lead with Latics going close through a Dieng strike from distance.
On the stroke of half time, Shrewsbury won a free kick 35 yards and after Coleman’s initial fumble, he recovered to clear the ball from danger.
Athletic made a bright start to the second half as David Dunn’s surging run and pass found the onrushing Jones to shoot but Jayson Leutwiler tipped over the crossbar.
Latics continued their onslaught and won a penalty on 51 minutes as Dunn was fouled in the penalty area.
The referee adjudged Ogogo to have brought down the former Blackburn Rovers man and Kelly stepped up to fire Latics into the lead, sending the goalkeeper the wrong way.
Straight from the goal, Latics went in search for more but Philliskirk’s cross fell straight into the goalkeepers arms after promising build up.
Shrewsbury made a double substitution, bringing on Jordan Clark and Jean-Louis Akpa Akpro and that seemed to tip the balance of the game.
Just after the hour mark they felt they should have been awarded a penalty after Akpa Akpro went down in the penalty area but the referee was unmoved.
The visitors’ pressure paid 20 minutes from time as they managed to get the equaliser.
Striker James Collins nicked the ball off the toe of Jonathan Burn, rounded Coleman and slid the ball into an empty goal to pull the Shrews level.
Latics were really struggling to hold onto the ball and Shrewsbury looked to capitalise on a vulnerable looking defence.
The visitors could have won the game with 15 minutes of the game remaining as Ellis’ header was blocked on the line by Jones whilst M.Woods’ follow up was well saved by Coleman.
With Latics defending for their lives, substitute George Green entered the referee’s notebook for a late challenge on a Shrewsbury midfielder.
Goalscorer Collins curled a shot over the bar from the edge of the box whilst at the other end, Kelly fired over from ten yards as Latics tried to find the winner.
Neither side could find the winning goal but Latics did extend their unbeaten run to the league season.
Oldham Athletic: Coleman, Brown, Burn, Wilson, Dieng, Kelly, Jones (Turner 90’), Dunn (Winchester 72’), Croft (Green 72’), Philliskirk, Forte.
Subs not used: Cornell, Edmundson, Poleon, Murphy.
Shrewsbury Town: Leutwiler, Tootle, Sadler, R.Woods, Grandison, M.Woods, Collins, Ellis, Ogogo (Whalley 69’), McAlinden (Clark 56’), Barnett (Akpa Akpro 56’).
Subs not used: Lawrence, Wesolowski, Halstead, Patterson.
Goals: Oldham Athletic (Kelly PEN 52’) 1-1 Shrewsbury Town (Collins 70’)
Attendance: 3,963 (364 visiting supporters)