The introduction of Conor Grimes was pivotal as Louth advanced to the next round of the All-Ireland SFC qualifiers.
Louth had a battle on their hands up to that but Grimes struck 1-1 after he was drafted in and his side pushed on from there.
Louth had a dreadful start to the year when they were relegated from Division 3 and they will be joining Leitrim in the lower tier next term.
But Leitrim have now won just once in 16 qualifiers and really never looked like changing in that at the Gaelic Grounds in Drogheda.
Louth were 0-11 to 0-8 in front after an entertaining first-half where both sides lit up the Gaelic Grounds with free-flowing football.
The home side started well with James Califf and Bevan Duffy giving them a 0-2 to 0-0 lead.
Kevin Conlon got Leitrim off the mark in the sixth minute but Louth restored their lead when Ronan Holdcroft landed a great score.
Louth corner-back Padraig Rath was given a black card in the ninth minute following a third-man tackle on Gary Reynolds, and Fergal Clancy brought Leitrim level with two quick scores.
Paraic Smith put back Louth back ahead and by the time Burns added his second they were three up.
However Clancy was on fire and hit two in a row up the other end to keep Leitrim right in contention.
Holdcroft kicked the point of the half off the outside of his boot before half-time, but with the last kick before the interval, Conlon made it 0-11 to 0-8, which kept Leitrim in touch.
Louth adapted to the conditions after break and Smith fired them further in front before Conlon kicked his second free but Louth hit the next two scores.
Colm Judge got the first, a point and then substitute Grimes struck the crucial goal in the 48th minute.
Danny Beck fired back for Leitrim but Grimes added a point and Declan Byrne pushed Louth clear.
Conlon slotted another free but it was in vain as Tommy Durnin and Derek Maguire finished it off for Louth.
Scores and teams
Scorers – Louth: C Grimes 1-1; J Califf 0-3 (0-3fs), R Burns 0-3 (1f), R Holdcroft 0-2, P Smith 0-2, C Judge 0-2; B Duffy 0-1, D Byrne 0-1, T Durnin 0-1, D Maguire 0-1.
Leitrim: K Conlon 0-5 (0-3fs), F Clancy 0-4; A McLoughlin 0-1, D Beck 0-1.
Louth: C Lynch; P Rath, D Finnegan, A Williams; D Byrne, D O'Hanlon, A Reid; T Durnin, J Califf; B Duffy, P Smith, C McKeever; R Burns, C Judge, R Holdcroft.
Subs: E Lafferty for Rath (10 mins, Black Card), D Maguire for McKeever (40) C Grimes for Burns (47) G McSorley for Smith (58) R Moore for Duffy (63).
Leitrim: C McCrann; M Murphy, R Gallagher, N Woods; S McWeeney (capt), G Reynolds, S Quinn; D Sweeney, S Moran; D Beck, P McGowan, K Conlan; F Clancy, D Wrynn, A McLoughlin.
Subs: J McGrail for Moran (21 mins), R Kennedy for McGowan (44), C Farrell for Beck (51), M Plunkett for Wrynn (58).
Referee: Ciaran Brannigan (Down).