Stephen O'Flynn headed in the opener in the 22nd minute after his initial penalty was saved by Marty Gallagher.
Michael McCrudden equalised from the penalty spot four minutes later.
John Currie's shot was deflected in off Mark Scoltock on 71 to give his side a lead ahead of Tuesday's second leg.
The Mallards ended up in the Premiership play-off position after losing 2-1 to Carrick Rangers on a dramatic final day of the regular season.
Whitey Anderson's side are aiming to ensure that they will play top-flight football for a fifth consecutive campaign, while Institute are hoping to return there after a one-year absence.
Institute took their place in the play-off despite finishing fifth in Championship One because H&W Welders, Armagh City and Knockbreda did not have Premiership licences.
O'Flynn dealt an early blow to his former club when he found the net at the third attempt after Gallagher saved his spot-kick.
McCrudden brought the sides level soon after, but Currie's effort gives his side the advantage going into the return leg at Ferney Park.
Ballinamallard United took a step towards preserving their Premiership status by beating Institute 2-1 in the first leg of their promotion-relegation play-off at the Riverside Stadium.