He scored after only five minutes but Ballymena led 10-7 at half-time thanks to Chris Henry's 27th-minute try and five points from the boot of scrum half Craig Woods.
Woods kicked 11 points in all but a late Jeremy Manning penalty saw the Leesiders hold onto the spoils.
The top flight's other Ulster club, Dungannon, remain in relegation trouble after they were edged out 23-19 by play-off hopefuls Clontarf at Stevenson Park.
'Gannon suffered their fourth defeat in five league games as Daragh O'Shea's late penalty confirmed the win for Clontarf.
'Tarf winger Derek Keane swapped first half tries with Dungannon flanker Thomas Anderson and a brace of penalties from Ian Keatley had the Dubliners 13-10 ahead at half-time.
'Gannon's Richard McCarter kept his side in the hunt with four penalties but a try from prop Kenny Dorian and 13 points between kickers O'Shea and Keatley proved crucial for the visitors.
Meanwhile, Garryowen continued their fine run of form by disposing of Galwegians by 20-7 at Crowley Park for their eighth league win on the bounce.
Tries in either half from second row duo Frank McKenna and Mark Melbourne kept the visitors' play-off push on track, with full-back Conor Kilroy contributing 10 points.
At Dr. Hickey Park, Greystones pulled themselves off the bottom of the table with a hard-fought 25-18 victory over UCD.
'Stones moved up to fifteenth as tries from Jamie Hagan, Stuart O'Connell and David McKechnie saw them overcome the students.
UCD did run in three unconverted tries, through Cian Aherne, John Lee and Vasily Artemiev, for a 15-12 interval lead.
However they could only add a Killian Lett penalty in the second half with full-back McKechnie's score providing the inspiration for the hosts.
Old Belvedere scored a bonus point 35-14 win over Lansdowne at Anglesea Road as they secured their Division One status for next season.
This was Lansdowne's fifth league defeat on the trot and it leaves them in serious trouble at the foot of the table.
The Ireland Under-20-capped Felix Jones and Eamonn O'Malley grabbed a try apiece, while Leinster's Richie Leyden and Tim Jago (2) also got on the scoresheet for 'Belvo.
Blackrock College moved up to sixth in the standings as a last-minute try from Ireland Under-20 scrum half David Moore saw them beat UL Bohemians 23-20 at the UL Sports Bowl.
Meanwhile, another number 9 had a day to remember at Templeville Road. Terenure's Mark Mahony grabbed a hat trick of tries to inspire his side to a 31-20 triumph over St. Mary's College.
Mahony crossed for two early opportunistic scores as 'Nure claimed a 10-0 lead in as many minutes.
Former Leinster out-half David McAllister booted a penalty and a drop goal before Mahony rounded off a fantastic team move in the 37th-minute for his third try.
Trailing 23-6 at the break, Mary's managed to make it a three-point game by the 70-minute mark thanks to converted tries from winger Daragh Fanning and scrum half Matthew D'Arcy.
But 'Nure, ending a club record nine league defeats in a row, found another gear with a second McAllister drop goal and an injury-time try from winger Cathal Morley.
DIVISION ONE: RESULTS -
Ballymena 16 Cork Constitution 18, Eaton Park
Scorers: Ballymena: Try: Chris Henry; Con: Craig Woods; Pens: Craig Woods 3
Cork Constitution: Tries: Richie Lane 2; Con: Jeremy Manning; Pens: Jeremy Manning 2
HT: Ballymena 10 Cork Constitution 7
Dungannon 19 Clontarf 23, Stevenson Park
Scorers: Dungannon: Try: Thomas Anderson; Con: Richard McCarter; Pens: Richard McCarter 4
Clontarf: Tries: Derek Keane, Kenny Dorian; Cons: Ian Keatley, Daragh Oâ00Shea; Pens: Ian Keatley 2, Daragh Oâ00Shea
HT: Dungannon 10 Clontarf 13
Galwegians 7 Garryowen 20, Crowley Park
Scorers: Galwegians: Try: Alipate Tuilevuka; Con: Liam Bibo
Garryowen: Tries: Frank McKenna, Mark Melbourne; Cons: Conor Kilroy 2; Pens: Conor Kilroy 2
HT: Galwegians 0 Garryowen 10
Greystones 25 UCD 18, Dr. Hickey Park
Scorers: Greystones: Tries: Jamie Hagan, Stuart Oâ00Connell, David McKechnie; Cons: James Power 2; Pens: James Power, Mark Geraghty
UCD: Tries: Cian Aherne, John Lee, Vasily Artemiev; Pen: Killian Lett
HT: Greystones 12 UCD 15
Old Belvedere 35 Lansdowne 14, Anglesea Road
Scorers: Old Belvedere: Tries: Tim Jago 2, Felix Jones, Richie Leyden, Eamonn Oâ00Malley; Cons: Richie Murphy 2, Brian Collins 3
Lansdowne: Try: Vinnie Oâ00Dowd; Pens: Andrew Geraghty 3
HT: Old Belvedere 14 Lansdowne 6
UL Bohemians 20 Blackrock College 23, UL Bowl
Scorers: UL Bohemians: Tries: Martin McPhail, Fergal Lawlor; Cons: Fergal Lawlor 2; Pens: Fergal Lawlor 2
Blackrock College: Tries: Brian Canavan, Paul Ryan, David Moore; Con: Kyle Tonetti; Pens: Fionn Carr, Kyle Tonetti
HT: UL Bohemians 7 Blackrock College 8
DIVISION TWO: RESULTS -
Ballynahinch 26 Old Crescent 32, Ballymacarn Park
Bective Rangers 33 Highfield 5, Donnybrook
Belfast Harlequins 14 Suttonians 6, Deramore Park
Clonakilty 22 Thomond 20, Shannonvale Cross
Malone 0 Buccaneers 24, Gibson Park
UCC 36 Barnhall 21, The Mardyke
Wanderers 25 Dublin University 16, Merrion Road
Young Munster 26 De La Salle Palmerston 7, Tom Clifford Park
DIVISION THREE: RESULTS -
Ards 15 City of Derry 25, Hamilton Park
County Carlow 10 Connemara 12, Oak Park
Corinthians 7 Instonians 20, Corinthian Park
Midleton 21 Banbridge 28, Towns Park
Naas 3 Bruff 34, Forenaughts
Nenagh Ormond 29 Rainey Old Boys 7, New Ormond Park
Queen's University 58 Sunday's Well 12, Dub Lane
Waterpark 3 Old Wesley 13, Ballinakill