He will be assisted by veteran former Aberdeen and St Mirren boss Alex Smith, who had been director of football under May.
May's departure after less than eight months in charge followed reports he had quit in the wake of Wednesday night's 1-0 Scottish Premier League defeat to Kilmarnock.
That left the club three points adrift at the foot of the table.
Pressley told the Bairns' official website: "Today has been a day of mixed emotions for me, with the disappointment of Eddie May leaving the club, but also the excitement of being given the opportunity to manage Falkirk Football Club.
"Working with Eddie over the last seven months has been a hugely enjoyable experience and we grew very close during that time.
"During Eddie's tenure here, he made many strong and decisive decisions and I'm sure that the club will benefit from these.
"There are many facets to being a football manager, whether it is tactical, motivational or psychological and although Eddie and I agree on some of these, our philosophies differed greatly on others.
"I have been given the opportunity to lead Falkirk to the end of the season and have a great belief in the players, backroom staff, the board and finally myself in securing Falkirk Football Club's SPL status.
"There are still 15 games remaining with a potential of 45 points to be won and, make no mistake, all 15 of these games are potentially winnable.
"I feel I have a good relationship with the players and I would not have taken on this job if I did not believe that I can manage them and bring the club a degree of success."
May's tenure began with an embarrassing exit from the Europa League qualifiers at the hands of Liechtenstein-based minnows FC Vaduz.
He had been on the coaching staff at Falkirk for five years before making the step up to his first senior management position.
After winning his first game in charge - the first leg of the Europa League second qualifying round back in July - May oversaw a 13-match winless run in league and cup.
That sequence finally came to an end with an SPL victory over Hamilton in November but the club continued to struggle, winning only one of their next 10 games.
They did record an impressive victory at Aberdeen just over a week ago but Wednesday night's result appears to have been the final straw.
May said: "It is with sadness that I will be leaving Falkirk Football Club today. I have thoroughly enjoyed my time at the club.
"Everyone knows the affection that I have for this football club, having been a player, academy director and latterly first-team manager.
"I would never have compromised this football club and I felt that it was best for everyone if I moved on.
"The Celtic match at Falkirk was a particular high for me as a manager, I was proud to manage the team which included nine academy graduates.
"Finally, I would like to thank the players and my staff for their outstanding support during my tenure, whilst I would also like to place on record my thanks to the directors for their backing during my time as academy director and for the opportunity to be manager of this club.
"In my time as a player and as manager, the fans have, on the whole, given me tremendous support and I thank them for that.
"This club will always hold a special place in my heart and I am sure Falkirk Football Club will go on to enjoy continued success in the Scottish Premier League."
Chairman Martin Ritchie said: "Eddie May has served this club extremely well for the last six years. In his role as academy director he has played a key role in transforming the club's youth system and developing our football infrastructure.
"He took on a very difficult and challenging task in the summer, leaving a secure position, and unfortunately that has not quite worked as we would all have hoped.
"I am sure that Eddie will go on to continue his career elsewhere in football and do extremely well.
"He has certainly left a very positive impression on Falkirk Football Club, particularly in the area of youth development.
"Everybody at the club would like to wish Eddie every success in the future."
May becomes the third SPL boss to lose his job since the turn of the year, following Jim Jefferies at Kilmarnock and Csaba Laszlo at Hearts.
Only four top-flight clubs - Rangers, Hamilton, St Mirren and St Johnstone - are now led by the managers who were in charge at the end of last season.
Hearts 8/11, Draw 5/2, Falkirk 4/1
Source: Team Talk