Premier League: Hull City 0 Arsenal 3
Ramsey, starting his first game since Boxing Day after overcoming a thigh injury, netted his ninth league goal of the season to give Arsenal the lead at the KC Stadium on Sunday before setting up Podolski to double his side's advantage before the break.
The Germany international then netted his fourth goal in two games early in the second half after Ramsey had been denied by Steve Harper.
Hull, who started the game brightly, struggled against a confident Arsenal team in the second half and could have gone further behind had Olivier Giroud and Per Mertesacker converted chances.
Arsenal's victory ensures Everton cannot overtake them this weekend. Hull, meanwhile, remain six points clear of the relegation zone with four games remaining.
Nikica Jelavic and Podolski went close after a quiet opening 15 minutes, but both were denied by good saves.
With 25 minutes played, Shane Long was guilty of wasting an excellent chance and the visitors made him pay six minutes later.
A crisp passing move culminated in Mesut Ozil, back after a hamstring injury, combining with Santi Cazorla to find Ramsey, and the Wales international found the net with a low finish.
Hull went agonisingly close to levelling four minutes later as Jake Livermore picked up a loose ball and unleashed a long-range strike that beat Wojciech Szczesny, but struck the post before bouncing to safety.
Arsenal doubled their lead in first-half stoppage time through Podolski, although a cluster of Hull players felt Jelavic had been fouled by Mikel Arteta in the build-up.
Podolski timed his run into the penalty area perfectly to lash home after Ramsey had cushioned Giroud's cross on his chest.
Arsene Wenger's men made it 3-0 nine minutes after the break. Cazorla teed up Ramsey on the edge of the area, his effort was saved by Harper with his feet, but the ball fell to Podolski and he calmly fired home from seven yards.
The visitors were very nearly gifted a fourth in the 64th minute as Harper misjudged a backpass from James Chester. Giroud blocked the goalkeeper's clearance and struck the crossbar from a tight angle.
Mertesacker was next to go close as he met a 75th-minute free-kick from Cazorla, but his looping header was just too high.
Ramsey missed a late chance to net his second, but it did not matter as Arsenal completed a comfortable win over their FA Cup final opponents and took another step towards securing UEFA Champions League football for next season.